We show that for rigid symmetric top molecules in electric fields the phenomenon of monodromy arises naturally as a "defect'' in the lattice of quantum states in the energy-momentum diagram. This makes it impossible to use either the total angular momentum or a pendular quantum number to label the states globally. The monodromy is created or destroyed by classical Hamiltonian Hopf bifurcations from relative equilibria. These phenomena are robust and should be observable in quasi-symmetric top molecules with field strengths epsilon satisfying muepsilon/b > 4.5, where mu is the dipole moment and b the rotational constant perpendicular to the symmetry axis of the molecule. (C) 2003 American Institute of Physics.