Does it? I think that could describe “associative” equally well as “commutative”.
Edit: “symmetric” might be an alternative for Abelian because the Cayley table (https://en.wikipedia.org/wiki/Cayley_table) of an Abelian group is symmetric. I’m not sure that’s immediately clear enough for laymen, though.
"Symmetric group" is already taken. Given a set S, the symmetric group G_s is the group of all permutations of S. Not to be confused with a permutation group of S, which is a group of some permutations of S.
I suppose we could rename "symmetric group" to "maximal permutation group", but stringing adjectives together is not a sustainable strategy for naming. Plus, it is not clear to me how accurate "maximal permutation group" is as a name. That is, the symmetric group S_3 is the maximal permutation group of {1,2,3}. However, it is not generally a maximal permutation group, as it is contained withing the permutation group S_4.
Symmetric is also good but it has a very distinct geometric interpretation. If we can avoid overloading terminology in a confusing way when designing these the names we should do so.
Perhaps pair-independent is a distinctive name for associative...
Well, introducing the new term “Abelian” certainly avoids overloading terminology in a confusing way :-)
“Pair invariant” IMO, isn’t good, certainly worse than “Pairing invariant”. “Parentheses invariant” might work, but of course would get just as confusing/incorrect as “pair invariant” once one moves from groups to fields.
"Order-independent" like I suggested communicates a lot.