  We undertook a mutually complementary analytic and computational study of the
full-fledged spherical (3D) quantum rotor subject to combined orienting and
aligning interactions characterized, respectively, by dimensionless parameters
$\eta$ and $\zeta$. By making use of supersymmetric quantum mechanics (SUSY
QM), we found two sets of conditions under which the problem of a spherical
quantum pendulum becomes analytically solvable. These conditions coincide with
the loci $\zeta=\frac{\eta^2}{4k^2}$ of the intersections of the eigenenergy
surfaces spanned by the $\eta$ and $\zeta$ parameters. The integer topological
index $k$ is independent of the eigenstate and thus of the projection quantum
number $m$. These findings have repercussions for rotational spectra and
dynamics of molecules subject to combined permanent and induced dipole
interactions.
