We undertook a mutually complementary analytic and computational study of the full-fledged spherical (three-dimensional) quantum rotor subject to combined orienting and aligning interactions, corresponding to a linear polar and polarizable molecule interacting with an electric field. The orienting and aligning interactions are characterized, respectively, by dimensionless parameters η and ζ. By making use of supersymmetric quantum mechanics, we found two sets of conditions (cases A and B) 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. In case A, the topological index k=∓m+1 with m the angular momentum projection quantum number, whereas, in case B, k=1, independent of m. These findings have repercussions for rotational spectra and dynamics of molecules subject to combined permanent and induced dipole interactions.