  We report on previously unobserved inter-system degeneracies in two-component
equal-mass Fermi gases with interspecies zero-range interactions under
isotropic harmonic confinement. Over the past 10 years, two-component Fermi
gases consisting of $n_1$ spin-up and $n_2$ spin-down atoms with interspecies
zero-range interactions have become a paradigm for modeling condensed matter
systems, nuclear matter and neutron matter. We show that the eigen energies of
the $(n_1+1,n_2-1)$ system are degenerate with the eigen energies of the
$(n_1,n_2)$ system for any s-wave scattering length $a_s$, including infinitely
large, positive and negative $a_s$. The existence of the inter-system
degeneracies is demonstrated explicitly for few-body systems with $n_1+n_2=4,
5$ and 6. The degeneracies and associated symmetries are explained within a
group theoretical framework.
