text:
Pairs of palindromic anagramic dates separated by a palindromic number of days
Definitions:
A palindrome is a string which reads the same backward or forward (not counting spaces or special characters), such as "madam" or "Sorel Eros".
A date is said to be a palindrome when its dd-mm-yyyy format is a palindrome (or any other local representation).
Two strings are said to be anagrams when the contain the same characters (modulo a rearrangement), such as "Silent" and "Listen". Two dates are anagrams when their representation (the same as the one chosen for 1.) are anagrams.
Challenge:
The challenge is to find all the pairs (a,b) of dates (between 1/1/01 and 12/31/9999) which satisfy the following three conditions :

a and b are different
a and b are palindromes
a and b are anagrams
a and b are separated by a number of days which is a palindrome.

Constraints
You can choose one of the three date formats :

yyyy-mm-dd
dd-mm-yyyy
mm-dd-yyyy

That means left padding with zeros for days and month (02/02/1999) and a 4 zeros padding for years.
You must respect leap years + every 100 years leap + every 400 years leap such as specified in the Proleptic Gregorian calendar
Input
None.
Output
Pairs of dates (same format) and the number of days separating them
Display format can be different to the computation format.
31-03-3013 - 13-03-3031 :: 6556
[EDIT]
Standard loopholes apply
This is code golf, shortest code wins!

Related: list all palindromic prime dates between 0000-01-01 and 99999-12-31

Are single digit numbers palindromes? if they are, we need to list all palindromic dates too. So, are a and b different dates?

When are the numbers left-padded with zeros and with how many zeroes?

Edited to address comments. Yes a and b are different and the numbers are padded (february -> 02, december -> 12 and same for day)

Haha just checking if both dates are palindromes for the first 50 years times out in TIO for PHP, gonna warm up the max_execution_time in preparation

Why having edited for "no zero padding for years"?, it wasn't asked for and will bother many I think.. can we have the choice on this?

Are we allowed to check the palindromizes for dates with format dd-mm-yyyy, yet still output them as format yyyy-mm-dd?

@KevinCruijssen yes; Computation must be in the same format system, but output can be any format

Polyglot, 0 bytes
When using the formats yyyy-mm-dd or mm-dd-yyyy, the list of palindrome dates are:
yyyy-mm-dd: [0101-10-10, 0110-01-10, 0111-11-10, 0120-02-10, 0121-12-10, 0130-03-10, 0140-04-10, 0150-05-10, 0160-06-10, 0170-07-10, 0180-08-10, 0190-09-10, 0201-10-20, 0210-01-20, 0211-11-20, 0220-02-20, 0221-12-20, 0230-03-20, 0240-04-20, 0250-05-20, 0260-06-20, 0270-07-20, 0280-08-20, 0290-09-20, 0301-10-30, 0310-01-30, 0311-11-30, 0321-12-30, 0330-03-30, 0340-04-30, 0350-05-30, 0360-06-30, 0370-07-30, 0380-08-30, 0390-09-30, 1001-10-01, 1010-01-01, 1011-11-01, 1020-02-01, 1021-12-01, 1030-03-01, 1040-04-01, 1050-05-01, 1060-06-01, 1070-07-01, 1080-08-01, 1090-09-01, 1101-10-11, 1110-01-11, 1111-11-11, 1120-02-11, 1121-12-11, 1130-03-11, 1140-04-11, 1150-05-11, 1160-06-11, 1170-07-11, 1180-08-11, 1190-09-11, 1201-10-21, 1210-01-21, 1211-11-21, 1220-02-21, 1221-12-21, 1230-03-21, 1240-04-21, 1250-05-21, 1260-06-21, 1270-07-21, 1280-08-21, 1290-09-21, 1301-10-31, 1310-01-31, 1321-12-31, 1330-03-31, 1350-05-31, 1370-07-31, 1380-08-31, 2001-10-02, 2010-01-02, 2011-11-02, 2020-02-02, 2021-12-02, 2030-03-02, 2040-04-02, 2050-05-02, 2060-06-02, 2070-07-02, 2080-08-02, 2090-09-02, 2101-10-12, 2110-01-12, 2111-11-12, 2120-02-12, 2121-12-12, 2130-03-12, 2140-04-12, 2150-05-12, 2160-06-12, 2170-07-12, 2180-08-12, 2190-09-12, 2201-10-22, 2210-01-22, 2211-11-22, 2220-02-22, 2221-12-22, 2230-03-22, 2240-04-22, 2250-05-22, 2260-06-22, 2270-07-22, 2280-08-22, 2290-09-22, 3001-10-03, 3010-01-03, 3011-11-03, 3020-02-03, 3021-12-03, 3030-03-03, 3040-04-03, 3050-05-03, 3060-06-03, 3070-07-03, 3080-08-03, 3090-09-03, 3101-10-13, 3110-01-13, 3111-11-13, 3120-02-13, 3121-12-13, 3130-03-13, 3140-04-13, 3150-05-13, 3160-06-13, 3170-07-13, 3180-08-13, 3190-09-13, 3201-10-23, 3210-01-23, 3211-11-23, 3220-02-23, 3221-12-23, 3230-03-23, 3240-04-23, 3250-05-23, 3260-06-23, 3270-07-23, 3280-08-23, 3290-09-23, 4001-10-04, 4010-01-04, 4011-11-04, 4020-02-04, 4021-12-04, 4030-03-04, 4040-04-04, 4050-05-04, 4060-06-04, 4070-07-04, 4080-08-04, 4090-09-04, 4101-10-14, 4110-01-14, 4111-11-14, 4120-02-14, 4121-12-14, 4130-03-14, 4140-04-14, 4150-05-14, 4160-06-14, 4170-07-14, 4180-08-14, 4190-09-14, 4201-10-24, 4210-01-24, 4211-11-24, 4220-02-24, 4221-12-24, 4230-03-24, 4240-04-24, 4250-05-24, 4260-06-24, 4270-07-24, 4280-08-24, 4290-09-24, 5001-10-05, 5010-01-05, 5011-11-05, 5020-02-05, 5021-12-05, 5030-03-05, 5040-04-05, 5050-05-05, 5060-06-05, 5070-07-05, 5080-08-05, 5090-09-05, 5101-10-15, 5110-01-15, 5111-11-15, 5120-02-15, 5121-12-15, 5130-03-15, 5140-04-15, 5150-05-15, 5160-06-15, 5170-07-15, 5180-08-15, 5190-09-15, 5201-10-25, 5210-01-25, 5211-11-25, 5220-02-25, 5221-12-25, 5230-03-25, 5240-04-25, 5250-05-25, 5260-06-25, 5270-07-25, 5280-08-25, 5290-09-25, 6001-10-06, 6010-01-06, 6011-11-06, 6020-02-06, 6021-12-06, 6030-03-06, 6040-04-06, 6050-05-06, 6060-06-06, 6070-07-06, 6080-08-06, 6090-09-06, 6101-10-16, 6110-01-16, 6111-11-16, 6120-02-16, 6121-12-16, 6130-03-16, 6140-04-16, 6150-05-16, 6160-06-16, 6170-07-16, 6180-08-16, 6190-09-16, 6201-10-26, 6210-01-26, 6211-11-26, 6220-02-26, 6221-12-26, 6230-03-26, 6240-04-26, 6250-05-26, 6260-06-26, 6270-07-26, 6280-08-26, 6290-09-26, 7001-10-07, 7010-01-07, 7011-11-07, 7020-02-07, 7021-12-07, 7030-03-07, 7040-04-07, 7050-05-07, 7060-06-07, 7070-07-07, 7080-08-07, 7090-09-07, 7101-10-17, 7110-01-17, 7111-11-17, 7120-02-17, 7121-12-17, 7130-03-17, 7140-04-17, 7150-05-17, 7160-06-17, 7170-07-17, 7180-08-17, 7190-09-17, 7201-10-27, 7210-01-27, 7211-11-27, 7220-02-27, 7221-12-27, 7230-03-27, 7240-04-27, 7250-05-27, 7260-06-27, 7270-07-27, 7280-08-27, 7290-09-27, 8001-10-08, 8010-01-08, 8011-11-08, 8020-02-08, 8021-12-08, 8030-03-08, 8040-04-08, 8050-05-08, 8060-06-08, 8070-07-08, 8080-08-08, 8090-09-08, 8101-10-18, 8110-01-18, 8111-11-18, 8120-02-18, 8121-12-18, 8130-03-18, 8140-04-18, 8150-05-18, 8160-06-18, 8170-07-18, 8180-08-18, 8190-09-18, 8201-10-28, 8210-01-28, 8211-11-28, 8220-02-28, 8221-12-28, 8230-03-28, 8240-04-28, 8250-05-28, 8260-06-28, 8270-07-28, 8280-08-28, 8290-09-28, 9001-10-09, 9010-01-09, 9011-11-09, 9020-02-09, 9021-12-09, 9030-03-09, 9040-04-09, 9050-05-09, 9060-06-09, 9070-07-09, 9080-08-09, 9090-09-09, 9101-10-19, 9110-01-19, 9111-11-19, 9120-02-19, 9121-12-19, 9130-03-19, 9140-04-19, 9150-05-19, 9160-06-19, 9170-07-19, 9180-08-19, 9190-09-19, 9201-10-29, 9210-01-29, 9211-11-29, 9220-02-29, 9221-12-29, 9230-03-29, 9240-04-29, 9250-05-29, 9260-06-29, 9270-07-29, 9280-08-29, 9290-09-29]
mm-dd-yyyy: [10-10-0101, 01-10-0110, 11-10-0111, 02-10-0120, 12-10-0121, 03-10-0130, 04-10-0140, 05-10-0150, 06-10-0160, 07-10-0170, 08-10-0180, 09-10-0190, 10-20-0201, 01-20-0210, 11-20-0211, 02-20-0220, 12-20-0221, 03-20-0230, 04-20-0240, 05-20-0250, 06-20-0260, 07-20-0270, 08-20-0280, 09-20-0290, 10-30-0301, 01-30-0310, 11-30-0311, 12-30-0321, 03-30-0330, 04-30-0340, 05-30-0350, 06-30-0360, 07-30-0370, 08-30-0380, 09-30-0390, 10-01-1001, 01-01-1010, 11-01-1011, 02-01-1020, 12-01-1021, 03-01-1030, 04-01-1040, 05-01-1050, 06-01-1060, 07-01-1070, 08-01-1080, 09-01-1090, 10-11-1101, 01-11-1110, 11-11-1111, 02-11-1120, 12-11-1121, 03-11-1130, 04-11-1140, 05-11-1150, 06-11-1160, 07-11-1170, 08-11-1180, 09-11-1190, 10-21-1201, 01-21-1210, 11-21-1211, 02-21-1220, 12-21-1221, 03-21-1230, 04-21-1240, 05-21-1250, 06-21-1260, 07-21-1270, 08-21-1280, 09-21-1290, 10-31-1301, 01-31-1310, 12-31-1321, 03-31-1330, 05-31-1350, 07-31-1370, 08-31-1380, 10-02-2001, 01-02-2010, 11-02-2011, 02-02-2020, 12-02-2021, 03-02-2030, 04-02-2040, 05-02-2050, 06-02-2060, 07-02-2070, 08-02-2080, 09-02-2090, 10-12-2101, 01-12-2110, 11-12-2111, 02-12-2120, 12-12-2121, 03-12-2130, 04-12-2140, 05-12-2150, 06-12-2160, 07-12-2170, 08-12-2180, 09-12-2190, 10-22-2201, 01-22-2210, 11-22-2211, 02-22-2220, 12-22-2221, 03-22-2230, 04-22-2240, 05-22-2250, 06-22-2260, 07-22-2270, 08-22-2280, 09-22-2290, 10-03-3001, 01-03-3010, 11-03-3011, 02-03-3020, 12-03-3021, 03-03-3030, 04-03-3040, 05-03-3050, 06-03-3060, 07-03-3070, 08-03-3080, 09-03-3090, 10-13-3101, 01-13-3110, 11-13-3111, 02-13-3120, 12-13-3121, 03-13-3130, 04-13-3140, 05-13-3150, 06-13-3160, 07-13-3170, 08-13-3180, 09-13-3190, 10-23-3201, 01-23-3210, 11-23-3211, 02-23-3220, 12-23-3221, 03-23-3230, 04-23-3240, 05-23-3250, 06-23-3260, 07-23-3270, 08-23-3280, 09-23-3290, 10-04-4001, 01-04-4010, 11-04-4011, 02-04-4020, 12-04-4021, 03-04-4030, 04-04-4040, 05-04-4050, 06-04-4060, 07-04-4070, 08-04-4080, 09-04-4090, 10-14-4101, 01-14-4110, 11-14-4111, 02-14-4120, 12-14-4121, 03-14-4130, 04-14-4140, 05-14-4150, 06-14-4160, 07-14-4170, 08-14-4180, 09-14-4190, 10-24-4201, 01-24-4210, 11-24-4211, 02-24-4220, 12-24-4221, 03-24-4230, 04-24-4240, 05-24-4250, 06-24-4260, 07-24-4270, 08-24-4280, 09-24-4290, 10-05-5001, 01-05-5010, 11-05-5011, 02-05-5020, 12-05-5021, 03-05-5030, 04-05-5040, 05-05-5050, 06-05-5060, 07-05-5070, 08-05-5080, 09-05-5090, 10-15-5101, 01-15-5110, 11-15-5111, 02-15-5120, 12-15-5121, 03-15-5130, 04-15-5140, 05-15-5150, 06-15-5160, 07-15-5170, 08-15-5180, 09-15-5190, 10-25-5201, 01-25-5210, 11-25-5211, 02-25-5220, 12-25-5221, 03-25-5230, 04-25-5240, 05-25-5250, 06-25-5260, 07-25-5270, 08-25-5280, 09-25-5290, 10-06-6001, 01-06-6010, 11-06-6011, 02-06-6020, 12-06-6021, 03-06-6030, 04-06-6040, 05-06-6050, 06-06-6060, 07-06-6070, 08-06-6080, 09-06-6090, 10-16-6101, 01-16-6110, 11-16-6111, 02-16-6120, 12-16-6121, 03-16-6130, 04-16-6140, 05-16-6150, 06-16-6160, 07-16-6170, 08-16-6180, 09-16-6190, 10-26-6201, 01-26-6210, 11-26-6211, 02-26-6220, 12-26-6221, 03-26-6230, 04-26-6240, 05-26-6250, 06-26-6260, 07-26-6270, 08-26-6280, 09-26-6290, 10-07-7001, 01-07-7010, 11-07-7011, 02-07-7020, 12-07-7021, 03-07-7030, 04-07-7040, 05-07-7050, 06-07-7060, 07-07-7070, 08-07-7080, 09-07-7090, 10-17-7101, 01-17-7110, 11-17-7111, 02-17-7120, 12-17-7121, 03-17-7130, 04-17-7140, 05-17-7150, 06-17-7160, 07-17-7170, 08-17-7180, 09-17-7190, 10-27-7201, 01-27-7210, 11-27-7211, 02-27-7220, 12-27-7221, 03-27-7230, 04-27-7240, 05-27-7250, 06-27-7260, 07-27-7270, 08-27-7280, 09-27-7290, 10-08-8001, 01-08-8010, 11-08-8011, 02-08-8020, 12-08-8021, 03-08-8030, 04-08-8040, 05-08-8050, 06-08-8060, 07-08-8070, 08-08-8080, 09-08-8090, 10-18-8101, 01-18-8110, 11-18-8111, 02-18-8120, 12-18-8121, 03-18-8130, 04-18-8140, 05-18-8150, 06-18-8160, 07-18-8170, 08-18-8180, 09-18-8190, 10-28-8201, 01-28-8210, 11-28-8211, 02-28-8220, 12-28-8221, 03-28-8230, 04-28-8240, 05-28-8250, 06-28-8260, 07-28-8270, 08-28-8280, 09-28-8290, 10-09-9001, 01-09-9010, 11-09-9011, 02-09-9020, 12-09-9021, 03-09-9030, 04-09-9040, 05-09-9050, 06-09-9060, 07-09-9070, 08-09-9080, 09-09-9090, 10-19-9101, 01-19-9110, 11-19-9111, 02-19-9120, 12-19-9121, 03-19-9130, 04-19-9140, 05-19-9150, 06-19-9160, 07-19-9170, 08-19-9180, 09-19-9190, 10-29-9201, 01-29-9210, 11-29-9211, 02-29-9220, 12-29-9221, 03-29-9230, 04-29-9240, 05-29-9250, 06-29-9260, 07-29-9270, 08-29-9280, 09-29-9290]

All possible date-pairs which are anagrams between one-another are:
yyyy-mm-dd: [[0101-10-10,0110-10-01], [0101-10-10,1001-01-10], [0101-10-10,1010-01-01], [0102-10-20,0120-10-02], [0102-10-20,1002-01-20], [0102-10-20,1020-01-02], [0102-10-20,2001-02-10], [0102-10-20,2010-02-01], [0103-10-30,0130-10-03], [0103-10-30,1003-01-30], [0103-10-30,1030-01-03], [0103-10-30,3001-03-10], [0103-10-30,3010-03-01], [0110-10-01,1001-01-10], [0110-10-01,1010-01-01], [0111-10-11,1011-01-11], [0111-10-11,1101-11-10], [0111-10-11,1110-11-01], [0112-10-21,0121-10-12], [0112-10-21,1012-01-21], [0112-10-21,1021-01-12], [0112-10-21,1102-11-20], [0112-10-21,1120-11-02], [0112-10-21,2011-02-11], [0112-10-21,2101-12-10], [0112-10-21,2110-12-01], [0113-10-31,0131-10-13], [0113-10-31,1013-01-31], [0113-10-31,1031-01-13], [0113-10-31,1103-11-30], [0113-10-31,1130-11-03], [0113-10-31,3011-03-11], [0120-10-02,1002-01-20], [0120-10-02,1020-01-02], [0120-10-02,2001-02-10], [0120-10-02,2010-02-01], [0121-10-12,1012-01-21], [0121-10-12,1021-01-12], [0121-10-12,1102-11-20], [0121-10-12,1120-11-02], [0121-10-12,2011-02-11], [0121-10-12,2101-12-10], [0121-10-12,2110-12-01], [0122-10-22,1022-01-22], [0122-10-22,2012-02-21], [0122-10-22,2021-02-12], [0122-10-22,2102-12-20], [0122-10-22,2120-12-02], [0130-10-03,1003-01-30], [0130-10-03,1030-01-03], [0130-10-03,3001-03-10], [0130-10-03,3010-03-01], [0131-10-13,1013-01-31], [0131-10-13,1031-01-13], [0131-10-13,1103-11-30], [0131-10-13,1130-11-03], [0131-10-13,3011-03-11], [0132-10-23,1032-01-23], [0132-10-23,2031-02-13], [0132-10-23,2103-12-30], [0132-10-23,2130-12-03], [0132-10-23,3012-03-21], [0132-10-23,3021-03-12], [0140-10-04,1040-01-04], [0140-10-04,4001-04-10], [0140-10-04,4010-04-01], [0141-10-14,1041-01-14], [0141-10-14,1140-11-04], [0141-10-14,4011-04-11], [0142-10-24,1042-01-24], [0142-10-24,2041-02-14], [0142-10-24,2140-12-04], [0142-10-24,4012-04-21], [0142-10-24,4021-04-12], [0150-10-05,1050-01-05], [0150-10-05,5001-05-10], [0150-10-05,5010-05-01], [0151-10-15,1051-01-15], [0151-10-15,1150-11-05], [0151-10-15,5011-05-11], [0152-10-25,1052-01-25], [0152-10-25,2051-02-15], [0152-10-25,2150-12-05], [0152-10-25,5012-05-21], [0152-10-25,5021-05-12], [0160-10-06,1060-01-06], [0160-10-06,6001-06-10], [0160-10-06,6010-06-01], [0161-10-16,1061-01-16], [0161-10-16,1160-11-06], [0161-10-16,6011-06-11], [0162-10-26,1062-01-26], [0162-10-26,2061-02-16], [0162-10-26,2160-12-06], [0162-10-26,6012-06-21], [0162-10-26,6021-06-12], [0170-10-07,1070-01-07], [0170-10-07,7001-07-10], [0170-10-07,7010-07-01], [0171-10-17,1071-01-17], [0171-10-17,1170-11-07], [0171-10-17,7011-07-11], [0172-10-27,1072-01-27], [0172-10-27,2071-02-17], [0172-10-27,2170-12-07], [0172-10-27,7012-07-21], [0172-10-27,7021-07-12], [0180-10-08,1080-01-08], [0180-10-08,8001-08-10], [0180-10-08,8010-08-01], [0181-10-18,1081-01-18], [0181-10-18,1180-11-08], [0181-10-18,8011-08-11], [0182-10-28,1082-01-28], [0182-10-28,2081-02-18], [0182-10-28,2180-12-08], [0182-10-28,8012-08-21], [0182-10-28,8021-08-12], [0190-10-09,1090-01-09], 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[07-11-1170,01-17-7110], [08-11-1180,11-08-8011], [08-11-1180,10-18-8101], [08-11-1180,01-18-8110], [09-11-1190,11-09-9011], [09-11-1190,10-19-9101], [09-11-1190,01-19-9110], [10-21-1201,01-21-1210], [10-21-1201,11-02-2011], [10-21-1201,10-12-2101], [10-21-1201,01-12-2110], [01-21-1210,11-02-2011], [01-21-1210,10-12-2101], [01-21-1210,01-12-2110], [11-21-1211,11-12-2111], [02-21-1220,12-02-2021], [02-21-1220,02-12-2120], [02-21-1220,10-22-2201], [02-21-1220,01-22-2210], [12-21-1221,12-12-2121], [12-21-1221,11-22-2211], [03-21-1230,03-12-2130], [03-21-1230,12-03-3021], [03-21-1230,02-13-3120], [03-21-1230,10-23-3201], [03-21-1230,01-23-3210], [04-21-1240,04-12-2140], [04-21-1240,12-04-4021], [04-21-1240,02-14-4120], [04-21-1240,10-24-4201], [04-21-1240,01-24-4210], [05-21-1250,05-12-2150], [05-21-1250,12-05-5021], [05-21-1250,02-15-5120], [05-21-1250,10-25-5201], [05-21-1250,01-25-5210], [06-21-1260,06-12-2160], [06-21-1260,12-06-6021], [06-21-1260,02-16-6120], [06-21-1260,10-26-6201], [06-21-1260,01-26-6210], [07-21-1270,07-12-2170], [07-21-1270,12-07-7021], [07-21-1270,02-17-7120], [07-21-1270,10-27-7201], [07-21-1270,01-27-7210], [08-21-1280,08-12-2180], [08-21-1280,12-08-8021], [08-21-1280,02-18-8120], [08-21-1280,10-28-8201], [08-21-1280,01-28-8210], [09-21-1290,09-12-2190], [09-21-1290,12-09-9021], [09-21-1290,02-19-9120], [09-21-1290,10-29-9201], [09-21-1290,01-29-9210], [10-31-1301,01-31-1310], [10-31-1301,11-03-3011], [10-31-1301,10-13-3101], [10-31-1301,01-13-3110], [01-31-1310,11-03-3011], [01-31-1310,10-13-3101], [01-31-1310,01-13-3110], [12-31-1321,12-13-3121], [12-31-1321,11-23-3211], [03-31-1330,03-13-3130], [05-31-1350,05-13-3150], [05-31-1350,03-15-5130], [07-31-1370,07-13-3170], [07-31-1370,03-17-7130], [08-31-1380,08-13-3180], [08-31-1380,03-18-8130], [10-02-2001,01-02-2010], [11-02-2011,10-12-2101], [11-02-2011,01-12-2110], [12-02-2021,02-12-2120], [12-02-2021,10-22-2201], [12-02-2021,01-22-2210], [03-02-2030,02-03-3020], [04-02-2040,02-04-4020], [05-02-2050,02-05-5020], [06-02-2060,02-06-6020], [07-02-2070,02-07-7020], [08-02-2080,02-08-8020], [09-02-2090,02-09-9020], [10-12-2101,01-12-2110], [02-12-2120,10-22-2201], [02-12-2120,01-22-2210], [12-12-2121,11-22-2211], [03-12-2130,12-03-3021], [03-12-2130,02-13-3120], [03-12-2130,10-23-3201], [03-12-2130,01-23-3210], [04-12-2140,12-04-4021], [04-12-2140,02-14-4120], [04-12-2140,10-24-4201], [04-12-2140,01-24-4210], [05-12-2150,12-05-5021], [05-12-2150,02-15-5120], [05-12-2150,10-25-5201], [05-12-2150,01-25-5210], [06-12-2160,12-06-6021], [06-12-2160,02-16-6120], [06-12-2160,10-26-6201], [06-12-2160,01-26-6210], [07-12-2170,12-07-7021], [07-12-2170,02-17-7120], [07-12-2170,10-27-7201], [07-12-2170,01-27-7210], [08-12-2180,12-08-8021], [08-12-2180,02-18-8120], [08-12-2180,10-28-8201], [08-12-2180,01-28-8210], [09-12-2190,12-09-9021], [09-12-2190,02-19-9120], [09-12-2190,10-29-9201], [09-12-2190,01-29-9210], [10-22-2201,01-22-2210], [03-22-2230,02-23-3220], [04-22-2240,02-24-4220], [05-22-2250,02-25-5220], [06-22-2260,02-26-6220], [07-22-2270,02-27-7220], [08-22-2280,02-28-8220], [09-22-2290,02-29-9220], [10-03-3001,01-03-3010], [11-03-3011,10-13-3101], [11-03-3011,01-13-3110], [12-03-3021,02-13-3120], [12-03-3021,10-23-3201], [12-03-3021,01-23-3210], [04-03-3040,03-04-4030], [05-03-3050,03-05-5030], [06-03-3060,03-06-6030], [07-03-3070,03-07-7030], [08-03-3080,03-08-8030], [09-03-3090,03-09-9030], [10-13-3101,01-13-3110], [02-13-3120,10-23-3201], [02-13-3120,01-23-3210], [12-13-3121,11-23-3211], [04-13-3140,03-14-4130], [05-13-3150,03-15-5130], [06-13-3160,03-16-6130], [07-13-3170,03-17-7130], [08-13-3180,03-18-8130], [09-13-3190,03-19-9130], [10-23-3201,01-23-3210], [04-23-3240,03-24-4230], [05-23-3250,03-25-5230], [06-23-3260,03-26-6230], [07-23-3270,03-27-7230], [08-23-3280,03-28-8230], [09-23-3290,03-29-9230], [10-04-4001,01-04-4010], [11-04-4011,10-14-4101], [11-04-4011,01-14-4110], [12-04-4021,02-14-4120], [12-04-4021,10-24-4201], [12-04-4021,01-24-4210], [05-04-4050,04-05-5040], [06-04-4060,04-06-6040], [07-04-4070,04-07-7040], [08-04-4080,04-08-8040], [09-04-4090,04-09-9040], [10-14-4101,01-14-4110], [02-14-4120,10-24-4201], [02-14-4120,01-24-4210], [12-14-4121,11-24-4211], [05-14-4150,04-15-5140], [06-14-4160,04-16-6140], [07-14-4170,04-17-7140], [08-14-4180,04-18-8140], [09-14-4190,04-19-9140], [10-24-4201,01-24-4210], [05-24-4250,04-25-5240], [06-24-4260,04-26-6240], [07-24-4270,04-27-7240], [08-24-4280,04-28-8240], [09-24-4290,04-29-9240], [10-05-5001,01-05-5010], [11-05-5011,10-15-5101], [11-05-5011,01-15-5110], [12-05-5021,02-15-5120], [12-05-5021,10-25-5201], [12-05-5021,01-25-5210], [06-05-5060,05-06-6050], [07-05-5070,05-07-7050], [08-05-5080,05-08-8050], [09-05-5090,05-09-9050], [10-15-5101,01-15-5110], [02-15-5120,10-25-5201], [02-15-5120,01-25-5210], [12-15-5121,11-25-5211], [06-15-5160,05-16-6150], [07-15-5170,05-17-7150], [08-15-5180,05-18-8150], [09-15-5190,05-19-9150], [10-25-5201,01-25-5210], [06-25-5260,05-26-6250], [07-25-5270,05-27-7250], [08-25-5280,05-28-8250], [09-25-5290,05-29-9250], [10-06-6001,01-06-6010], [11-06-6011,10-16-6101], [11-06-6011,01-16-6110], [12-06-6021,02-16-6120], [12-06-6021,10-26-6201], [12-06-6021,01-26-6210], [07-06-6070,06-07-7060], [08-06-6080,06-08-8060], [09-06-6090,06-09-9060], [10-16-6101,01-16-6110], [02-16-6120,10-26-6201], [02-16-6120,01-26-6210], [12-16-6121,11-26-6211], [07-16-6170,06-17-7160], [08-16-6180,06-18-8160], [09-16-6190,06-19-9160], [10-26-6201,01-26-6210], [07-26-6270,06-27-7260], [08-26-6280,06-28-8260], [09-26-6290,06-29-9260], [10-07-7001,01-07-7010], [11-07-7011,10-17-7101], [11-07-7011,01-17-7110], [12-07-7021,02-17-7120], [12-07-7021,10-27-7201], [12-07-7021,01-27-7210], [08-07-7080,07-08-8070], [09-07-7090,07-09-9070], [10-17-7101,01-17-7110], [02-17-7120,10-27-7201], [02-17-7120,01-27-7210], [12-17-7121,11-27-7211], [08-17-7180,07-18-8170], [09-17-7190,07-19-9170], [10-27-7201,01-27-7210], [08-27-7280,07-28-8270], [09-27-7290,07-29-9270], [10-08-8001,01-08-8010], [11-08-8011,10-18-8101], [11-08-8011,01-18-8110], [12-08-8021,02-18-8120], [12-08-8021,10-28-8201], [12-08-8021,01-28-8210], [09-08-8090,08-09-9080], [10-18-8101,01-18-8110], [02-18-8120,10-28-8201], [02-18-8120,01-28-8210], [12-18-8121,11-28-8211], [09-18-8190,08-19-9180], [10-28-8201,01-28-8210], [09-28-8290,08-29-9280], [10-09-9001,01-09-9010], [11-09-9011,10-19-9101], [11-09-9011,01-19-9110], [12-09-9021,02-19-9120], [12-09-9021,10-29-9201], [12-09-9021,01-29-9210], [10-19-9101,01-19-9110], [02-19-9120,10-29-9201], [02-19-9120,01-29-9210], [12-19-9121,11-29-9211], [10-29-9201,01-29-9210]]

The differences in days of those pairs are:
yyyy-mm-dd: [3278, 328445, 331723, 6557, 328445, 335001, 693344, 696622, 9835, 328445, 338280, 1058239, 1061517, 325167, 328445, 328445, 361620, 364898, 3278, 328444, 331723, 361619, 368176, 693343, 726517, 729795, 6556, 328445, 335001, 361619, 371454, 1058238, 321888, 328444, 686787, 690065, 325166, 328445, 358341, 364898, 690065, 723239, 726517, 328445, 690065, 693344, 723239, 729796, 318610, 328445, 1048404, 1051682, 321889, 328445, 355063, 364898, 1051682, 328444, 693343, 719960, 729795, 1051682, 1054960, 328444, 1410024, 1413302, 328445, 364898, 1413302, 328445, 693344, 729796, 1413303, 1416581, 328445, 1771643, 1774921, 328445, 364898, 1774921, 328444, 693343, 729795, 1774921, 1778199, 328444, 2133263, 2136541, 328445, 364898, 2136541, 328445, 693344, 729796, 2136542, 2139820, 328445, 2494882, 2498160, 328445, 364898, 2498160, 328444, 693343, 729795, 2498160, 2501438, 328444, 2856502, 2859780, 328445, 364898, 2859780, 328445, 693344, 729796, 2859781, 2863059, 328445, 3218122, 3221400, 328445, 364898, 3221400, 328444, 693343, 729795, 3221400, 3224678, 3278, 6556, 364899, 368177, 9835, 729794, 733072, 33175, 36453, 3279, 33175, 39732, 364899, 398073, 401351, 6556, 33174, 43009, 729793, 358343, 361621, 29896, 36453, 361620, 394794, 398072, 361620, 364899, 394794, 401351, 719959, 723237, 26618, 36453, 723237, 364899, 391516, 401351, 723238, 726516, 1081580, 1084858, 36453, 1084857, 364899, 401351, 1084858, 1088136, 1443198, 1446476, 36453, 1446476, 364899, 401351, 1446477, 1449755, 1804819, 1808097, 36453, 1808096, 364899, 401351, 1808097, 1811375, 2166437, 2169715, 36453, 2169715, 364899, 401351, 2169716, 2172994, 2528058, 2531336, 36453, 2531335, 364899, 401351, 2531336, 2534614, 2889677, 2892955, 36453, 2892955, 364899, 401351, 2892956, 2896234, 3278, 6557, 331724, 364898, 368176, 9835, 696619, 3278, 364897, 325167, 358341, 361619, 361619, 361620, 364898, 686784, 358341, 364897, 1048404, 364898, 1410023, 364897, 1771643, 364898, 2133262, 364897, 2494882, 364898, 2856502, 364897, 3278, 6556, 33174, 36452, 3279, 33174, 39731, 29895, 36452, 355060, 361617, 26617, 36452, 358339, 361617, 361617, 716681, 723238, 36452, 719959, 723237, 723237, 1078299, 1084856, 36452, 1081578, 1084856, 1084856, 1439920, 1446477, 36452, 1443198, 1446476, 1446476, 1801538, 1808095, 36452, 1804817, 1808095, 1808095, 2163159, 2169716, 36452, 2166437, 2169715, 2169715, 2524778, 2531335, 36452, 2528057, 2531335, 2531335, 3278, 6557, 9835, 331722, 335000, 3278, 6556, 321887, 325165, 683507, 686785, 1045126, 1048404, 1406746, 1410024, 1768365, 1771643, 2129985, 2133263, 2491605, 2494883, 3278, 6557, 9835, 3278, 6556, 351785, 361620, 361620, 361621, 713404, 723239, 716683, 723239, 723239, 1075024, 1084859, 1084859, 1084860, 1436643, 1446478, 1439922, 1446478, 1446478, 1798263, 1808098, 1801542, 1808098, 1808099, 2159883, 2169718, 2169718, 2169718, 3278, 6557, 9835, 3278, 361619, 361619, 361618, 723239, 723239, 723239, 1084858, 1084858, 1084857, 1446478, 1446478, 1446478, 1808098, 1808098, 1808097, 3278, 6557, 9835, 3278, 6556, 361620, 361620, 361621, 723239, 723239, 723239, 1084859, 1084859, 1084860, 1446479, 1446479, 1446479, 3278, 6557, 9835, 3278, 361619, 361619, 361618, 723239, 723239, 723239, 1084859, 1084859, 1084858, 3278, 6557, 9835, 3278, 6556, 361620, 361620, 361621, 723240, 723240, 723240, 3278, 6557, 9835, 3278, 6556, 361620, 361620, 361619, 3278, 6557, 9835, 3278]
mm-dd-yyyy: [3014, 328709, 331723, 325695, 328709, 328709, 361560, 364574, 29837, 32851, 328709, 687256, 690270, 32851, 328709, 364574, 394412, 397426, 690270, 723121, 726135, 62690, 65704, 328709, 1048818, 1051832, 328709, 1410378, 1413392, 328709, 1771939, 1774953, 328709, 2133499, 2136513, 328709, 2495060, 2498074, 328709, 2856620, 2859634, 328709, 3218180, 3221194, 3014, 298872, 657419, 660433, 295858, 654405, 657419, 295858, 331723, 361561, 364575, 657419, 690270, 693284, 657419, 364575, 657419, 693284, 723121, 726135, 657419, 1018981, 657419, 1380541, 657419, 1742102, 657419, 2103662, 657419, 2465223, 657419, 2826783, 657419, 3188343, 3014, 266019, 986128, 989142, 263005, 983114, 986128, 298870, 361561, 364575, 986128, 1018979, 1021993, 331722, 660431, 986128, 1021993, 1051831, 1054845, 986128, 986128, 1347688, 986128, 1709249, 986128, 2070809, 986128, 2432370, 986128, 2793930, 986128, 3155490, 3014, 32851, 35865, 358547, 361561, 35865, 65703, 68717, 361561, 394412, 397426, 720109, 723123, 1081669, 1084683, 1443230, 1446244, 1804790, 1807804, 2166351, 2169365, 2527911, 2530925, 2889471, 2892485, 3014, 29838, 32852, 325696, 358547, 361561, 32852, 361561, 62691, 65705, 687258, 720109, 723123, 1048818, 1081669, 1084683, 1410379, 1443230, 1446244, 1771939, 1804790, 1807804, 2133500, 2166351, 2169365, 2495060, 2527911, 2530925, 2856620, 2889471, 2892485, 3014, 295858, 328709, 331723, 292844, 325695, 328709, 328709, 292844, 328709, 358546, 361560, 328709, 361560, 328709, 654406, 690271, 720109, 723123, 328709, 1015966, 1051831, 1081668, 1084682, 328709, 1377527, 1413392, 1443230, 1446244, 328709, 1739087, 1774952, 1804789, 1807803, 328709, 2100648, 2136513, 2166351, 2169365, 328709, 2462208, 2498073, 2527910, 2530924, 328709, 2823768, 2859633, 2889471, 2892485, 3014, 624567, 657418, 660432, 621553, 654404, 657418, 657418, 690270, 657418, 657418, 1380539, 657418, 2103660, 657418, 2465220, 3014, 32851, 35865, 35865, 65702, 68716, 361562, 723122, 1084683, 1446243, 1807804, 2169364, 2530924, 3014, 29837, 32851, 32851, 325697, 361562, 391400, 394414, 687257, 723122, 752959, 755973, 1048818, 1084683, 1114521, 1117535, 1410378, 1446243, 1476080, 1479094, 1771939, 1807804, 1837642, 1840656, 2133499, 2169364, 2199201, 2202215, 2495059, 2530924, 2560762, 2563776, 3014, 361563, 723122, 1084684, 1446243, 1807805, 2169364, 2530925, 3014, 32851, 35865, 35865, 65703, 68717, 361560, 723121, 1084681, 1446242, 1807802, 2169362, 3014, 29838, 32852, 32852, 361560, 723121, 1084681, 1446242, 1807802, 2169362, 3014, 361559, 723121, 1084680, 1446242, 1807801, 2169362, 3014, 32851, 35865, 35865, 65702, 68716, 361561, 723121, 1084682, 1446242, 1807802, 3014, 29837, 32851, 32851, 361561, 723121, 1084682, 1446242, 1807802, 3014, 361562, 723121, 1084683, 1446242, 1807803, 3014, 32851, 35865, 35865, 65703, 68717, 361560, 723121, 1084681, 1446241, 3014, 29838, 32852, 32852, 361560, 723121, 1084681, 1446241, 3014, 361559, 723121, 1084680, 1446241, 3014, 32851, 35865, 35865, 65702, 68716, 361561, 723121, 1084681, 3014, 29837, 32851, 32851, 361561, 723121, 1084681, 3014, 361562, 723121, 1084682, 3014, 32851, 35865, 35865, 65703, 68717, 361560, 723120, 3014, 29838, 32852, 32852, 361560, 723120, 3014, 361559, 723120, 3014, 32851, 35865, 35865, 65702, 68716, 361560, 3014, 29837, 32851, 32851, 361560, 3014, 361561, 3014, 32851, 35865, 35865, 65703, 68717, 3014, 29838, 32852, 32852, 3014]

And none of those differences are palindromes, so there won't be any output for either of these two formats.
(See my Java answer for an answer using format dd-mm-yyyy like the challenge description, which I've also used to generate the lists above.)

I found a similar solution to your Polyglot one in Golfscript - I'm going to see if I can make it work in Python next!

@Mathgeek Polyglot means it works in all languages, including Golfscript and Python. ;)

I'm aware, [I'm being intentionally dense for comedic value] ;p

Doesn't this violate a couple of the the standard loopholes like this or maybe this

@Xcali, we also have a consensus somewhere that proofs of "impossibility" are valid.

@Xcali Shaggy is indeed correct: Is a proof of impossibility allowed as an answer? Although that meta question is mainly directed to challenges with no valid solutions at all, so I guess there is a fine line in this challenge where alternative formats do give proper outputs.

Red, 305 bytes
p: func[s n][at to""10 ** n + s 2]s: func[d][rejoin[p d/4 2 p d/3 2 p d/2 4]]
r: func[s][s: to""s s = reverse copy s]a: to-date[1 1 1]b:
collect[until[if r s a[keep a]a: a + 1 a/2 = 1e4]]repeat i l: length? b[repeat
j l[if all[i < j(sort s b/:i)= sort s b/:j
r t: b/:j - b/:i][print[s b/:i s b/:j"::"t]]]]

Try it online!
In dd-mm-yyyy format. I'm listing the pairs only once. Currently there is no separator in the dates - I'll fix it if it's necessary.

JavaScript (ES6),  115 109  108 bytes
Date format: dd-mm-yyyy.
_=>[-1102,-1101,-2,0,2,4,5,7,9].map(y=>([,a,b,c,d]=y+11303+'',[a+b+(m='-'+c+d+'-'+d+c)+b+a,b+a+m+a+b,6556]))

Try it online!

Well done but 31-09-9013 does not exist in the current calendar

@RegisPortalez My bad. Now fixed.

Uses format ddmmyyyy for the palindrome checks like the challenge description.
Java 8 (hard-coded), 190 bytes
v->"31100113 13100131x20011002 02011020x31011013 13011031x20022002 02022020x31122113 13122131x31033013 13033031x31055013 13055031x31077013 13077031x31088013 13088031x".replace("x"," 6556\n")

As expected, a lot shorter than calculating it..
Outputs in the format ddmmyyyy.
Try it online.
Explanation:
v->   // Method with empty unused parameter and String return-type
  "31100113 13100131x20011002 02011020x31011013 13011031x20022002 02022020x31122113 13122131x31033013 13033031x31055013 13055031x31077013 13077031x31088013 13088031x"
      //  Return this String
    .replace("x"," 6556\n")
      //  after we've replaced all "x" with " 6556\n"

Java 10 (calculated), 524 486 476 bytes
import java.time.*;v->{var D=LocalDate.of(1,1,1);java.util.Arrays A=null;var L=D.datesUntil(D.of(10000,1,1)).filter(d->{var s=(d+"").split("-");return p(s[2]+s[1]+s[0]);}).toArray(LocalDate[]::new);for(var a:L)for(var b:L){var x=(a+"").split("");var y=(b+"").split("");A.sort(x);A.sort(y);var d=a.until(b,java.time.temporal.ChronoUnit.DAYS);if(d>0&A.equals(x,y)&p(d+""))System.out.println(a+" "+b+" "+d);}}boolean p(String s){return s.contains(new StringBuffer(s).reverse());}

Uses the same format as in the challenge description for the checks ddmmyyyy, but outputs as format yyyy-mm-dd.
Try it online.
Explanation:
import java.time.*;       // Import for both LocalDate
v->{                      // Method with empty unused parameter and no return-type
  var D=LocalDate.of(1,1,1);
                          //  LocalDate variable to save bytes, with date 01-01-0001
  java.util.Arrays A=null;//  Static Arrays to save bytes
  var L=D.datesUntil(D.of(10000,1,1))
                          //  Create a stream of dates in the range [01-01-0001,01-01-9999]:
        .filter(d->{      //  Filter it by:
          var s=(d+"")    //   Convert the date to a String in the format "yyyy-mm-dd"
            .split("-");  //   And split it at the "-"
      return p(s[2]+s[1]+s[0]);})
                          //   Return whether "ddmmyyyy" is a palindrome
        .toArray(LocalDate[]::new);
                          //  And convert that filtered stream to an array of LocalDates
for(var a:L)for(var b:L){ //  Nested loop over all these dates a,b:
  var x=(a+"")            //   Convert date `a` to a String,
        .split("");       //   and then into an array of characters
  var y=(b+"").split(""); //   Do the same for date `b`
  A.sort(x);A.sort(y);    //   Sort the characters in both arrays
  var d=a.until(b,java.time.temporal.ChronoUnit.DAYS);
                          //   Get the difference in days between the two dates
  if(d>0                  //   If this difference is larger than 0,
                          //   (which means a is before b, and they are not the same)
     &A.equals(x,y)       //   and the two sorted character-arrays are equal,
                          //   (which means a and b are anagrams)
     &p(d+""))            //   and the difference is a palindrome:
    System.out.println(a+" "+b+" "+d);}}
                          //    Print the pair a,b and difference with trailing newline

// Separated method to check if a String is a palindrome:
boolean p(String s){      // Method with String parameter and boolean return-type
  return s.contains(      //  Return whether the input-string contains:
    new StringBuffer(s).reverse());}
                          //   The reversed input-String(Buffer)

[HARCODED]
PHP, 194 bytes
<?=str_replace("x"," 6556
","31100113 13100131x20011002 02011020x31011013 13011031x20022002 02022020x31122113 13122131x31033013 13033031x31055013 13055031x31077013 13077031x31088013 13088031x");

Try it online!
[WITH CAUTION]: shorter version that assumes following Kevin Cruijssen's results (I haven't verified them) that the difference in days will always be 6556:
PHP, 223 bytes
$a=date_create('0000-12-31');for(;($c=($d=clone$a->modify('+1day'))->format('dmY'))!=31129999;)if($c==strrev($c)&($e=$d->modify('+6556day')->format('dmY'))==strrev($e)&count_chars($c,1)==count_chars($e,1))echo"$c $e 6556
";

Try it online!
[FULL VERSION THAT CALCULATES WITH ALL DATES]:
PHP, 283 bytes
$a=date_create('0000-12-31');for(;($c=($d=clone$a->modify('+1day'))->format('dmY'))!=31129999;){for(;($e=$d->modify('+1day')->format('dmY'))!=31129999;)if($c==strrev($c)&$e==strrev($e)&count_chars($c,1)==count_chars($e,1)&($f=$d->diff($a)->format('%a'))==strrev($f))echo"$c $e $f
";}

Try it online!
Dates format is dd-mm-yyyy, pairs are displayed in one order only. formats the inputs like dd-mm-yyyy dd-mm-yyyy n but doesn't separate them (count 1 bytes more for a new line).
Actually times out pretty quick.. cannot even check the first result in TIO. Note that the date 31-12-9999 is never tested (loops pre-increment), but it doesn't matter because it's not a palindrome anyway..
EDIT: added hardcoded version (port of Kevin Cruijssen's) + switched to format ddmmyyyy (without -) with a new line for the other versions

05AB1E, 39 bytes
31 9и20₂S;ǝDí«•1/K†Þ'āD•2×2ôøJº2äøŽP¶δª

05AB1E doesn't even have any date buitlins, so hard-coded it is. ;)
Uses and outputs in format ddmmyyyy. Outputs as a list of triplets.
Try it online (the » in the footer is to pretty-print them, but feel free to remove it to see the actual output).
Explanation:
31             # Push 31
   9и          # Repeat it 9 times as list: [31,31,31,31,31,31,31,31,31]
     20        # Push 20
       ₂S      # Push builtin 26 as digit list: [2,6]
         ;     # Halve each: [1,3]
          ǝ    # Insert 20 at both those (0-based) indices: [31,20,31,20,31,31,31,31,31]
           D   # Duplicate this list
            í  # Reverse each: [13,"02",13,"02",13,13,13,13,13]
             « # Merge the two lists together:
               #  [31,20,31,20,31,31,31,31,31,13,"02",13,"02",13,13,13,13,13]
•1/K†Þ'āD•    '# Push compressed integer 100101021203050708
          2×   # Double this as string: "100101021203050708100101021203050708"
            2ô # Split it into parts of size 2:
               #  [10,"01","01","02",12,"03","05","07","08",10,"01","01","02",12,"03","05","07","08"]
ø              # Zip the two lists together, creating pairs:
               #  [[31,10],[20,"01"],[31,"01"],[20,"02"],[31,12],[31,"03"],...]
 J             # Join each pair together to a string: [3110,2001,3101,2002,3112,3103,...]
  º            # Mirror each: [31100113,20011002,31011013,20022002,31122113,31033013,...]
   2ä          # Split the list into two equal parts (of size 9)
     ø         # Zip/transpose it, swapping rows/columns:
               #   [[31100113,13100131],[20011002,"02011020"],[31011013,13011031],...]
      ŽP¶      # Push compressed integer 6556
         δª    # And append it to each pair to create triplets
               # (after which that list of triplets is output implicitly as result)

See this 05AB1E tip of mine (section How to compress large integers?) to understand why •1/K†Þ'āD• is 100101021203050708 and ŽP¶ is 6556.


text:
Square-Cube Digit Usage
The point of this challenge is to find the smallest positive integer that uses up at least all of the digits supplied in the input after it is squared and cubed.
So, when provided an input such as 0123456789 (i.e. a number that finds the result):
69² = 4761
69³ = 328509

It means that 69 is the result of such an input. (Fun fact: 69 is the smallest number that uses up all of the decimal digits of 0 to 9 after squaring and cubing.)
Specification
The input doesn't have to be unique. For example, the input can be 1466 and here's the result:
4² = 16
4³ = 64

That means we can't just fulfill this input by just doing a number that only uses 1 digit of 6, it has to have 2 digits of 6 in the output.
Test cases
Here's an exhaustive list of all numbers from 1 to 10000.
1333 (or 3133/3313/3331) -> 111

Input specification

You may take input as a list of digits.
Since the test cases (somehow) have a bug in it, here's an extra rule: the 0's in the input will be ignored.

I went "Huh?" when I first saw it, but I think I understand.

Everything before "the point of this challenge" is extremely unclear. Also using 1 as an example input is probably the worst possible choice to make your explanation clear.

Will the input always be sorted from smallest to largest?

Can you explain what happens when there are zeros in the input. For example with an input of 50 your list says the output should be 5 which gives us the digits 25 125 which contains no zeros. Does this mean we can ignore zeros in the input? because your example with 69 as an output did not imply that.

@LevelRiverSt I think that list was created with a buggy implementation that didn't check for the square being zero when the cube was zero. My answer doesn't have that issue.

So what's the answer to the question? can we ignore zeros in the input or not? I note that 69 is the smallest number that works for an input of 123456789 whether the 0 is required or not.

Japt -æ, 17 16 15 14 11 bytes
Still very not happy with this! A little happier! Now I'm happy!
Takes input as an integer.
²+U³s)á dèN

Try it
²+U³s)á dèN     :Implicit map of each U in the range [0,input)
²               :U squared
 +              :Concatenate
  U³            :  U cubed
    s           :  Converted to a string (preventing the + from adding the 2 numbers)
     )          :End concatenate
      á         :All permutations
        d       :Any truthy (not 0) when
         è      :  Counting the occurrences of
          N     :    The array of inputs, which is implicitly cast to a string
                :Implicit output of first U to return true

05AB1E, 14 12 13 bytes
-2 bytes thanks to @KevinCruijssen
+1 byte thanks to @Grimmy
∞.Δ23SmJœIÅ?Z

Try it online!

Explanation
∞.Δ                  - First number that...
   23Sm              - Power of 2 and 3 [n^2, n^3]
       J             - Concatenated
        œ            - Permutations of this number
         IÅ?Z        - any of these start with the number

By the way it's quite slow...

Yup, looks good now!

Thanks @Grimmy not used the Å? much and think I got my a and b mixed up

Python 3, 78 bytes
f=lambda s,n=1:n*all(f'{n*n}{n**3}'.count(i)>=s.count(i)for i in s)or f(s,n+1)

Try it online!

Ah, very nice use of f-string. Just posted to see you're a few steps ahead! T_T

Python 2, 82 \$\cdots\$78 77 bytes
Added 2 bytes to fix an error kindly pointed out by S.S. Anne.
Switched to Python 2 thanks to Grimmy.
Saved a byte thanks to Arnauld!!!
f=lambda s,i=1:i*all((`i*i`+`i**3`).count(c)/s.count(c)for c in s)or f(s,i+1)

Try it online!

Jelly, 14 bytes
1*2,3DFœ&Ƒ@ʋ1#

Try it online!
A monadic link taking a list of digits and returning an integer in a single element list.
1          ʋ1# | Start with 1 and find the first integer where the following is true, using the input digit list as the right argument:
 *2,3          | - To the power of 2 and 3
     D         | - Convert to lists of decimal digits
      F        | - Flatten
       œ&Ƒ@    | - Check whether the inout digit list is invariant when intersected with this list of digits

JavaScript (ES7),  77  72 bytes
Takes input as a list of digits.
f=(a,k)=>([...[k*k]+k**3].sort()+'').match(a.sort().join`.*`)?k:f(a,-~k)

Try it online!
Commented
f = (                     // f is a recursive function taking:
  a,                      //   a[] = input
  k                       //   k   = counter, initially undefined
) =>                      //
  ( [...[k * k] + k ** 3] // concatenate k² and k³ and split the resulting string
    .sort()               // sort from lowest to highest digit
    + ''                  // coerce back to a string (this puts commas between the
                          // digits, but they are harmless)
  ).match(                // test whether it matches:
    a.sort()              //   the input list sorted the same way
    .join`.*`             //   joined with .* patterns, so that unused digits and
                          //   commas are ignored
  ) ?                     // if it does:
    k                     //   stop recursion and return k
  :                       // else:
    f(a, -~k)             //   try again with k + 1

Wolfram Language (Mathematica), 89 bytes
1//.t_/;ContainsNone[Subsets[Join@@IntegerDigits[t^{2,3}],Length@#],Permutations@#]:>t+1&

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R, 142 134 121 bytes
i=1;x=table(scan());l=function(t)x>table(strsplit(paste0(t^2,t^3),"")[[1]])[names(x)];while(any(l(i),is.na(l(i))))i=i+1;i

Try it online!
R, 152 146 144 130 bytes
(This is only if we have to be wrong like your test cases)

i=1;x=scan();x=table(x[x>0]);l=function(t)x>table(strsplit(paste0(t^2,t^3),"")[[1]])[names(x)];while(any(l(i),is.na(l(i))))i=i+1;i

Try it online!

Ruby, 67 bytes
Uses Arnauld's regex. Takes a list of digits. If that isn't allowed, add 6 bytes to change d.sort to d.chars.sort.
->d,i=0{i+=1until"#{i*i}#{i**3}".chars.sort*''=~/#{d.sort*'.*'}/;i}

Try it online!

Burlesque, 28 bytes
r0{{2 3}?^im}]mj{j\\z?}j+]fi

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r0       # Range from [0,inf]
{
 {2 3}?^ # {squared, cubed}
 im      # Concatenate
}]m      # Map over each and parse to string
j        # Swap stack
{
 j       # Swap
 \\      # List difference
 z?      # Is null
}
j+]      # Prepend input to make {input j \\ z?}
fi       # Find index s.t.

Perl 5 -pF, 70 bytes
$p=join'.*',sort@F;1while(join'',sort((++$\**2 .$\**3)=~/./g))!~/$p/}{

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Charcoal, 27 bytes
≔⁰ηＷΦχ›№θＩκ№⁺ＩＸη³×ηηＩκ≦⊕ηＩη

Try it online! Link is to verbose version of code. Explanation:
≔⁰η

Start at zero.
ＷΦχ

Repeat until none of the 10 digits satisfies...
›№θＩκ№⁺ＩＸη³×ηηＩκ

... the count of that digit in the input is greater than the count in the cube and the square...
≦⊕η

... increment the result.
Ｉη

Output the result.

C (gcc), 134 bytes
s,c,i,m;f(n){for(c=s=i=m=1;m;c=(s=++i*i)*i){int a[10]={};for(;c;s/=10)a[c%10]++,s&&a[s%10]++,c/=10;for(m=n;m*a[m%10]--;m/=10);}n=--i;}

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Ungolfed and with better variable names:
int square, cube, result, input_copy;

int f(int input)
{
    for(cube = square = result = input_copy = 1;
        input_copy;
        cube = (square = ++result * result) * result)
    {
        int digits[10] = { };
        for(; cube; square /= 10)
        {
            digits[cube % 10]++;

            if(square)
                digits[square % 10]++;

            cube /= 10;
        }
        for(input_copy = input;
            input_copy && digits[input_copy % 10]--;
            input_copy /= 10);
    }
    return result-1;
}

Haskell, 62 bytes
import Data.List
f l=[n|n<-[0..],[]==l\\(show=<<[n^2,n^3])]!!0

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Even with the import, Haskell's list difference function \\ is quite powerful.

C (gcc),  93  92 bytes
-1 Byte thanks to ceilingcat
D;I;C(N){N=N?C(N/10)+(N%10==D):0;}F(N){for(D=I=0;D<10;C(N)>C(I*I)+C(I*I*I)?D=!++I:++D);I=I;}

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ungolfed:
int D;
int I;

int CNT(N)  // count occurrences of digit D in number N
{
    if (N)
        return CNT(N / 10) + (N % 10 == D);
    return 0;
}

int F(N)
{
    for (D = I = 0; D < 10; ++I)
    {
        for (D = 0; D < 10; ++D)
            if (CNT(N) > CNT(I * I) + CNT(I * I * I))
                break;
    }

    return I-1;
}

@ceilingcat that's clever thanks

Why do you always have to outgolf me? And how do you do it?

source: Stack Exchange

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

sort: votes


text:
Branch Out - Monkey Edition
The Monkey has swung home to find his tree all in pieces.
He likes order and scoops up all his 'tree-parts' and sets them out in 3 groups.
The Monkey's 3 nested lists:
STEP = [['S', '1', 'B', '3'], ['S', '3', 'B', '11'], ['S', '5', 'B', '12'], ['S', '4', 'B', '13'], ['S', '2', 'B', '14']]

TRANSITION = [['T', '2', 'B', '4'], ['T', '7', 'B', '4'], ['T', '3', 'S', '4'], ['T', '5', 'S', '5'], ['T', '1', 'S', '2'], ['T', '8', 'S', '2'], ['T', '6', 'S', '1'], ['T', '9', 'S', '2'], ['T', '4', 'S', '1'], ['T', '10', 'S', '1']]

BRANCH = [['B', '3', 'T', '1'], ['B', '3', 'T', '7'], ['B', '4', 'S', '3'], ['B', '11', 'T', '3'], ['B', '11', 'T', '5'], ['B', '12', 'T', '6'], ['B', '12', 'T', '8'], ['B', '13', 'T', '4'], ['B', '13', 'T', '9'], ['B', '14', 'T', '2'], ['B', '14', 'T', '10']]

Each element holds information as such:
# Example

STEP[0]  =  ['S', '1', 'B', '3']

Where:

'S' is the STEP type
'1' is the STEP number id
'B' is the linked BRANCH type
'3' is the linked BRANCH number id

Starting from a STEP the data is all linked, so using the linked reference you can find the next element and the next until another STEP is reached.
This is some parameters of the data:

STEPS are connected to single BRANCHES
BRANCHES are connected to one or more TRANSITIONS
TRANSITIONS can be connected to a single BRANCH or STEP

The BRANCH data can have a fork where a single BRANCH id has one or more options for TRANSITIONS.
Re-create this monkey's tree (nested list format is fine) in the shortest way possible (he is not a fan of long instructions):
           [S1]
            |
          B3|_________
            |         |
           [T1]     [T7]
            |         |
            |       B4|
            |         |
           [S2]     [S3]

And list all the routes on this tree:
Routes = [[S1, B3, T1, S2], [S1, B3, T7, B4, S3]]

UPDATE: This is Code-golf and from the data given you need to output the display of the tree ( This can be different from my example but requires the same elements to be displayed in any format) and as you cycle through the problem to output all the possible routes as a BONUS.
Notes:

This is a single-rooted tree
You may use a different format for input but please explicitly state the input format in your solution

Pretty new here, still getting the etiquette down. Yes Code-golf and from the data given you need to output the display of the tree ( This can be different from my example but requires the same elements to be displayed in any format) and as you cycle through the problem to output all the possible routes as a BONUS.

Welcome to Code Golf! What are acceptable outputs for the tree? Can it just be nested lists? Or does it need to be some ascii-art? (in which case this tag should be added and the format should be specified some more)

@arnauld - thank you! It can just be nested lists - okay, fantastic Ill update that

Welcome to CCPG! I enjoy graph theory related challenges. Some questions: 1. The input seems a bit laborious; can we simply take a single list of tuples (x,y) where x and y are strings? E.g.: [('S1', 'B3'), ('B3', 'T1'), ('B3, 'T7'), ...]? 2. Can we assume that the graph described by the input is a single rooted tree?

Hi @ChasBrown -- Yes that will be fine, I thought that keeping them separated makes it easier to crunch when numbers are larger than 9 - but feel free to modify the input format -- as long as it can reproduce the tree and the input is explicitly stated in your solution - yes this is a single-rooted tree. Thank you for the feedback - I'll take it on board for future posts!

Java (JDK), 570 bytes
t->{return "Routes = "+Arrays.deepToString(Arrays.stream(r(t.get(0).get(0),t,t.get(0).get(0)).split("\\[")).filter(d->d.endsWith("]")).map(d->d.replaceAll("]", "")).map(d->d.split("(?=[A-Z])")).collect(Collectors.toList()).toArray());};String r(String s, List<List<String>>t, String b){List<String>o = t.stream().filter(e->e.get(0).equals(s)).map(e->e.get(1)).collect(Collectors.toList());if(s.matches("^S(?!(?:1)$)\\d+"))return s;if(o.size()>1){String x="";for(String n:o){x+="["+b+r(n,t,b+n)+"]";}return x;}if(o.size()==1)return s+r(o.get(0),t,b+o.get(0));return "]";}

As you may notice by my code I´m still learning a lot every day.
My answer is neighter pretty nor short and I honestly even hope this works for different testcases even though I tested a couple.
But I hope that this question gets the attention it deserves and there will be more answers I can learn from.
INPUT
String[][] STEP = { { "S1", "B3" },  ... ,{ "B14", "T2" }, { "B14", "T10" } };
List<List<String>> tupels = Arrays.stream(STEP).map(Arrays::asList).collect(Collectors.toList());
z(tupels);

I converted your given Input into an 2D-Java-Array String[][] and converted that into a List<List<String>>.
CODE
The first method calls a recursive method to get all routes and formats the output.
t->{
return "Routes ="+Arrays.deepToString(
  Arrays.stream(r(t.get(0).get(0),t,t.get(0).get(0)).split("\\["))
 .filter(d->d.endsWith("]")).map(d->d.replaceAll("]", ""))
 .map(d->d.split("(?=[A-Z])")).collect(Collectors.toList()).toArray());
};

The second method is the recursive method which gathers all the possible routes
String r(String s, List<List<String>>t, String b){
  List<String>o =t.stream().filter(e->e.get(0).equals(s)).map(e->e.get(1))
  .collect(Collectors.toList());
  if(s.matches("^S(?!(?:1)$)\\d+"))
    return s;
  if(o.size()>1){
    String x="";
    for(String n:o){
      x+="["+b+r(n,t,b+n)+"]";
    }
    return x;
  }
  if(o.size()==1)
    return s+r(o.get(0),t,b+o.get(0));
  return "]";
}

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all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/