  Odd-angular momentum exciton states are dark to light in monolayers of
transition-metal dichalcogenides and can be addressed only by two-photon
probes. Besides, $2p$ excitons states are expected to show a fine splitting
that arises from the peculiar electronic band structure of the material,
characterized by a finite Berry curvature around each valley. In this work we
study in detail the coupling of photons to $p$ exciton states and we find that
the Rashba spin-orbit interaction or a Skyrmion in the transition-metal
dichalcogenide substrate can be exploited to engineer finite optical selection
rules. The basis mechanism relies on a matching of the exciton angular momentum
with the winding of the Rashba spin-orbit interaction or the Skyrmion
topological charge. In a photonic cavity, the coupling is enhanced due to
photon confinement and the resulting polaritonic branches acquire a mixing with
$2p^\pm$ excitons, thus providing a useful tool to detect exciton
fine-splitting and and enable novel uses of odd-parity dark excitons.
