  We formulate a quantum phase space for molecular rotational and nuclear-spin
states. Taking in molecular geometry and nuclear-spin data, our framework
yields admissible position and momentum states, inter-convertible via a
generalized Fourier transform. We classify molecules into three types --
asymmetric, rotationally symmetric, and perrotationally symmetric -- with the
last type having no macroscopic analogue due to nuclear-spin statistics
constraints. We identify two features in perrotationally symmetric state spaces
that are Hamiltonian-independent and induced solely by symmetry and spin
statistics. First, many molecular species are intrinsically rotation-spin
entangled in a way that cannot be broken without transitioning to another
species or breaking symmetry. Second, each molecular position state houses an
internal pseudo-spin or "fiber" degree of freedom, and the fiber's Berry phase
or matrix after adiabatic changes in position yields naturally robust
operations, akin to braiding anyonic quasiparticles or realizing fault-tolerant
quantum gates. We outline scenarios where these features can be experimentally
probed.
