  In this paper we study the controllability problem for a symmetric-top
molecule, both for its classical and quantum rotational dynamics. The molecule
is controlled through three orthogonal electric fields interacting with its
electric dipole. We characterize the controllability in terms of the dipole
position: when it lies along the symmetry axis of the molecule neither the
classical nor the quantum dynamics are controllable, due to the presence of a
conserved quantity, the third component of the total angular momentum; when it
lies in the orthogonal plane to the symmetry axis, a quantum symmetry arises,
due to the superposition of symmetric states, which has no classical
counterpart. If the dipole is neither along the symmetry axis nor orthogonal to
it, controllability for the classical dynamics and approximate controllability
for the quantum dynamics are proved to hold. The approximate controllability of
the symmetric-top Schroedinger equation is established by using a Lie-Galerkin
method, based on block-wise approximations of the infinite-dimensional systems.
