  We consider an ensemble of homonuclear diatomic molecules coupled to the two
polarization directions of a Fabry-P\'erot cavity via fully quantum
simulations. Accompanied by analytical results, we identify a coupling
mechanism mediated simultaneously by the two perpendicular polarizations, and
inducing polaritonic relaxation towards molecular rotations. This mechanism is
related to the concept of light-induced conical intersections (LICI). However,
unlike LICIs, these non-adiabatic pathways are of collective nature, since they
depend on the \emph{relative} intermolecular orientation of all electronic
transition dipoles in the polarization plane. Notably, this rotational
mechanism directly couples the bright upper and lower polaritonic states, and
it stays in direct competition with the collective relaxation towards
dark-states. Our simulations indicate that the molecular rotational dynamics in
gas-phase cavity-coupled systems can serve as a novel probe for non-radiative
polaritonic decay towards the dark-states manifold.
