  Orbital degrees of freedom play an essential role in metals, semiconductors,
and strongly confined electronic systems. Experiments with ultracold atoms have
used highly anisotropic confinement to explore low-dimensional physics, but
typically eliminate orbital degrees of freedom by preparing motional ground
states in strongly confined directions. Here we prepare multi-band systems of
spin-polarized fermionic potassium ($^{40}$K) in the quasi-one-dimensional
(q1D) regime and quantify the strength of atom-atom correlations using
radio-frequency spectroscopy. The activation of orbital degrees of freedom
leads to a new phenomenon: a low-energy scattering channel that has even
particle-exchange parity along the q1D axis, as if the underlying interactions
were s-wave. This emergent exchange symmetry is enabled by orbital singlet wave
functions in the strongly confined directions, which also confer high-momentum
components to low-energy q1D collisions. We measure both the q1D odd-wave and
even-wave "contact" parameters for the first time, and compare them to
theoretical predictions of one-dimensional many-body models. The strength and
spatial symmetry of interactions are tuned by a p-wave Feshbach resonance and
by transverse confinement strength. Near resonance, the even-wave contact
approaches its theoretical unitary value, whereas the maximum observed odd-wave
contact remains several orders of magnitude below its unitary limit. Low-energy
scattering channels of multi-orbital systems, such as those found here, may
provide new routes for the exploration of universal many-body phenomena.
