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 semiclassical limit. Different asymmetric molecules such as the water molecule are taken as an 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.