  The strong interactions among nucleons have an approximate spin-isospin
exchange symmetry that arises from the properties of quantum chromodynamics in
the limit of many colors, $N_c$. However this large-$N_c$ symmetry is well
hidden and reveals itself only when averaging over intrinsic spin orientations.
Furthermore, the symmetry is obscured unless the momentum resolution scale is
close to an optimal scale that we call $\Lambda_{{\rm large-}N_c}$. We show
that the large-$N_c$ derivation requires a momentum resolution scale of
$\Lambda_{{\rm large-}N_c} \sim 500$ MeV. We derive a set of spin-isospin
exchange sum rules and discuss implications for the spectrum of $^{30}$P and
applications to nuclear forces, nuclear structure calculations, and
three-nucleon interactions.
