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 η and ζ. 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 ζ = η2 / (4k2) of the intersections of the eigenenergy surfaces spanned by the η and ζ 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.