Does LOGLOG = NLOGLOG? Define LOGLOG as the class of languages which can be computed in space O(loglog n) by a deterministic Turing machine (with two-way access to the input). Similarly define NLOGLOG as the class of languages which can be computed in space O(log log n) by a non-deterministic Turing machine (with two-way access to the input). Is it really not known that these classes differ?

I could only find some older surveys and a theorem that if they equal then L=NL (which is not just a trivial padding argument!), but somehow I feel that separating these classes cannot be that hard. Of course I might be completely wrong, but if every second bit of the input is the numbers from 1 to n in increasing order in binary, separated by some symbols, then the machines can already learn loglog n and with every other second bit we can input a problem that can fool a deterministic machine but not a non-deterministic one. I don't see yet exactly how this could be done but feels like a possible approach, as with this trick we can basically input a depth log n binary tree along with its structure instead of the usual linear tape.
The entry in the complexity zoo is surprisingly detailed; it claims that 
NLOGLOG = co-NLOGLOG in the paper 


  Nondeterministic computations in sublogarithmic space and space constructibility, Viliam Geffert, SIAM Journal on Computing, 1991.


But after a brief reading, I do not see any claim about the fact that NLOGLOG is closed under complement; maybe a deeper look is needed.
And the main result they have is that there are no nondeterministic fully space-constructible unbounded monotone increasing $s(n)$ functions for $s(n) = o(\log n)$. It is known that if such functions exists, then 

$\mathsf{SPACE}[s(n)] \neq \mathsf{NSPACE}[s(n)]$. 

And in the conclusion the author claimed that " ...this main separation problem remains open. "

As @chazisop said, the relations of these low-level complexity classes are depended on the models, and it is stated in the entry of the zoo that


  "There are several possible definitions of this class; the most common is the class of languages which can be computed in space O(log log n) by a deterministic Turing machine with two-way access to the input. "


Which is coincident to your definition and also the paper's.