  We investigate the scattering properties and bound states of a
quasi-two-dimensional (q2D) spin-polarized Fermi gas near a $p$-wave Feshbach
resonance. Strong confinement promotes the out-of-plane spatial wave functions
to a discrete, gapped orbital degree of freedom. Exchange-antisymmetric orbital
pair wave functions are predicted to give rise to low-energy q2D interactions
with $s$-wave symmetry. Using radiofrequency (rf) spectroscopy, we observe the
signature power-law scaling and the dimensional-crossover feature anticipated
for the emergent $s$-wave channel. Additionally, we demonstrate that two types
of low-energy dimers, with either $s$-wave and $p$-wave symmetry, could be
formed via rf spin-flip association from an orbital mixture. These findings
illustrate how gapped orbital degrees of freedom can provide additional control
over scattering symmetries in strongly confined ultracold gases.
