Huh. I heard (US high school algebra in a recent unspecified century) the fact that ab=ba called "the commutative property of multiplication", and the "symmetric property of equality" meant a=b <=> b=a, and didn't run across anything later on (comp sci from a department of the engineering college rather than the math department) inconsistent with that. Symmetry sounds like a reasonable term for both, but it sounds like there's a distinction I don't get?