In this paper we study the controllability problem for a symmetric-top molecule, for both 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 Schriidinger equation is established by using a Lie-Galerkin method based on blockwise approximations of the infinite-dimensional systems.