  Nonadiabatic phenomena are investigated in the rovibrational motion of
molecules confined in an infrared cavity. Conical intersections (CIs) between
vibrational polaritons, similar to CIs between electronic polaritonic surfaces,
are found. The spectral, topological, and dynamic properties of the vibrational
polaritons show clear fingerprints of nonadiabatic couplings between molecular
vibration, rotation and the cavity photonic mode. Furthermore, it is found that
for the investigated system, composed of two rovibrating HCl molecules and the
cavity mode, breaking the molecular permutational symmetry, by changing
$^{35}$Cl to $^{37}$Cl in one of the HCl molecules, the polaritonic surfaces,
nonadiabatic couplings, and related spectral, topological, and dynamic
properties can deviate substantially. This implies that the natural occurrence
of different molecular isotopologues needs to be considered when modeling
realistic polaritonic systems.
