A standard method to detect thermal neutrons is the nuclear interaction $^3$He(n,p)$^3$H. The spin-dependence of this interaction is also the basis of a neutron spin-polarization filter using nuclear polarized $^3$He. We consider the corresponding interaction for neutrons placed in an intrinsic orbital angular momentum (OAM) state. We derive the relative polarization-dependent absorption cross-sections for neutrons in an $L=1$ OAM state. The absorption of those neutrons results in compound states $J^pi=0^-$, $1^-$, and $2^-$. Varying the three available polarizations tests that an OAM neutron has been absorbed and probes which decay states are physically possible. We describe the energetically likely excited states of $^4$He after absorption, due to the fact that the compound state has odd parity. This provides a definitive method for detecting neutron OAM states and suggests that intrinsic OAM states offer the possibility to observe new physics, including anomalous cross-sections and new channels of radioactive decay.
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