When is it legal to castle moving the rook first? As a follow on from the question Is it legal to castle with a "ghost" rook: When is it legal (according to the FIDE Laws of Chess) to castle moving the rook first?
Unless this is a trick question, I'd say never:

3.8.2 [] This is a move of the king and either rook of the same colour along the player’s first rank, counting as a single move of the king and executed as follows: the king is transferred from its original square two squares towards the rook on its original square, then that rook is transferred to the square the king has just crossed.

The word then implies the king moves first, and the rook second. Also,

4.4 If a player having the move:
4.4.1 touches his king and a rook he must castle on that side if it is legal to do so
4.4.2 deliberately touches a rook and then his king he is not allowed to castle on that side on that move and the situation shall be governed by Article 4.3.1

That would imply that touching both king and rook (but not moving them) commits the player to castling; then, he/she could then move the rook first and only then the king, since that would give the same position on the board as the only legal play at that moment. Still, if I would see that as an arbiter, I'd have a word with the player about this.