  We calculate the low-lying spectrum of the $^{16}$O nucleus using an
$\alpha$-cluster model which includes the important tetrahedral and square
configurations. Our approach is motivated by the dynamics of $\alpha$-particle
scattering in the Skyrme model. We are able to replicate the large energy
splitting that is observed between states of identical spin but opposite
parities, as well as introduce states that were previously not found in other
cluster models, such as a $0^-$ state. We also provide a novel interpretation
of the first excited state of $^{16}$O and make predictions for the energies of
$6^-$ states that have yet to be observed experimentally.
