[hope-striker]

What would you do with that?

For example, I can see the use of commutativity (ab = ba) and anticommutativity (ab = -ba), but I'm not sure what I'd do with the negation of commutativity (ab ≠ ba).

[pfortuny]

Nope: the negation is "there is a couple a,b such that ab!=ba", which means just "strictly not commutative group": I do not think there is a relevant theory to be done about them (otherwise, I guess it would have been done).

[hope-striker]

Ah, whoops. That seems even more useless, though.

[mikhailfranco]

Non-commutative also means that if there are units, there may be different left and right units:

    1L * A = A = A * 1R