the sign is a homomorphism" — one ℤ/2 invariant realized identically as permutation parity, a determinant sign, and a Legendre symbol

Zolotarev's lemma identifies the Legendre symbol with the sign of a permutation — reciprocity falling out of the fact that sign is a homomorphism to ℤ/2 that several structures share. The home for the move "this ℤ/2 invariant is one homomorphism in three costumes," and exactly the small case the group-action-residue shape ([[EX-05]]) generalizes: a residue that is a single sign because sign is a homomorphism.