  We prove an inverse spectral result for $S^1$-invariant metrics on $S^2$
based on the so-called asymptotic equivariant spectrum. This is roughly the
spectrum together with large weights of the $S^1$ action on the eigenspaces.
Our result generalizes an inverse spectral result of the first and last named
authors, together with Victor Guillemin, concerning $S^1$-invariant metrics on
$S^2$ which are invariant under the antipodal map. We use higher order terms in
the asymptotic expansion of a natural spectral measure associated with the
Laplacian and the $S^1$ action.
