The predicate is indeed symmetric:

?- random_permutation([1,2,3,4],L).
L = [3,4,1,2].

?- random_permutation(L,[3,4,1,2]).
L = [4,3,1,2].

But it does not accept a pair of permutations. For that, use permutation/2

?- random_permutation([1,2,3,4],[3,4,1,2]).
false.

?- permutation([1,2,3,4],[3,4,1,2]).
true ;
false.

A better predicate would be:

random_permutation(List, Permutation, PicklistFromListToPermutation).

where PicklistFromListToPermutation would the list of indexes picked from the original List. This means no information loss and a proper "reversal" would be possible.

For example:

random_permutation([1,2,3,4],[3,4,1,2],[2,3,0,1]).

Note to self: write that predicate