  We study both the classical and quantum rotational dynamics of an asymmetric
top molecule, controlled through three orthogonal electric fields that interact
with its dipole moment. The main difficulties in studying the controllability
of these infinite-dimensional quantum systems are the presence of severe
spectral degeneracies in the drift Hamiltonian and the nonsolvability of the
stationary free Schr\"odinger equation, which lead us to apply a perturbative
Lie algebraic approach.
  In this paper we show that, while the classical equations given by the
Hamiltonian system on ${\rm SO}(3)\times \mathbb{R}^3$ are controllable for all
values of the rotational constants and all dipole configurations, the
Sch\"odinger equation for the quantum evolution on $L^2({\rm SO}(3))$ is
approximately controllable for almost all values of the rotational constants if
and only if the dipole is not parallel to any of the principal axes of inertia
of the asymmetric rigid body.
