> Indeed. The word for a == b iff b == a is reflexive.
The answer is "both". The equality _relation_ is reflexive. The equality _operation_ is commutative.
> Commutative means that, for functions A:x->x, B:x->x: A(B(x)) == B(A(x)).
This doesn't fit any definition of "commutative" I was able to find. Every case I could locate involved the order of _operands_ to a single _binary_ function. Of course, you can turn you example into something like that, though with slightly different types, using higher-order functions (in pseudo-Haskell):
f1, f2 :: (a -> b) -> b
f1 = \f -> f A
f2 = \f -> f B
f1 (f2 (==)) == f2 (f1 (==))