Instead of thinking of a commutative function as a function that takes an ordered pair of inputs, we can think of as a function that takes an unordered bag of inputs, and therefore can't depend on their order. On this interpretation, the fact that functions are always given inputs in a particular order is an artifact of our definitions, not a fundamental property of functions themselves. If we had notation for functions applied to arguments in no particular order, then commutative functions would be the norm, and non-commutative functions would require additional structure imposed on their inputs.
In a world of linear left-to-right notation, where means " applied to first and second", commutativity looks like a constraint. In an alternative world where functions are applied to their inputs in parallel, with none of them distinguished as "first" by default, commutativity is the natural state of affairs.