  We study the classical and quantum rotation numbers of the free rotation of
asymmetric top molecules. We show numerically that the quantum rotation number
converges to its classical analog in the semi-classical limit. Different
asymmetric molecules such as the water molecule are taken as illustrative
example. A simple approximation of the classical rotation number is derived in
a neighborhood of the separatrix connecting the two unstable fixed points of
the system. Furthermore, a signature of the classical tennis racket effect in
the spectrum of asymmetric molecules is identified.
