"Order-independent" like I suggested communicates a lot.
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.
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.
Perhaps pair-independent is a distinctive name for associative...
“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.
https://dictionary.cambridge.org/dictionary/english/commute doesn’t give any definition close to the mathematical meaning, either.
Unless dictionaries 150-ish years ago had significantly closer descriptions of the term, “Abelian”, being a new term not loaded with pre-existing definition might be the better choice for naming this property.