Invariants for maps on the sphere

Question

From the Hopf theorem, we know that the degree classifies maps between spheres up to homotopy. Are more refined invariants (or useful notions of equivalence) that could distinguish between homotopic maps (and hence of the same degree) known?
For example, the degree of the identity and antipodal maps on $S^n$ are the same for $n$ odd, would any known (integer or algebraic) invariant distinguish between them?