  In 2007, McNamara proved that two skew shapes can have the same Schur support
only if they have the same number of $k\times \ell$ rectangles as subdiagrams.
This implies that two ribbons can have the same Schur support only if one is
obtained by permuting row lengths of the other. We present substantial progress
towards classifying when a permutation $\pi \in S_m$ of row lengths of a ribbon
$\alpha$ produces a ribbon $\alpha_{\pi}$ with the same Schur support as
$\alpha$; when this occurs for all $\pi \in S_m$, we say that $\alpha$ has
"full equivalence class." Our main results include a sufficient condition for a
ribbon $\alpha$ to have full equivalence class. Additionally, we prove a
separate necessary condition, which we conjecture to be sufficient.
