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.

"Associative" doesn't provide any information to avoid confusion when dealing with mixed operators either.

Somehow including "peer" in the terminology could help.