We report rigorous quantum five-dimensional (SD) calculations of the translation-rotation (T-R) energy levels and wave functions of H-2 inside aza-thia-open-cage fullerene (ATOCF). Translational and rotational excitations of this endohedral complex have been measured in a recent inelastic neutron scattering (INS) study, enabling direct comparison between theory and experiment. ATOCF has no symmetry, and therefore the intermolecular potential energy surface (PES) governing the T-R dynamics of H-2 is strongly anisotropic. A pairwise additive PES is employed in the calculations. Inspection of the wave functions shows three regular quasi-ID translational modes aligned with the Cartesian x, y, and z axes, respectively. These and other translational excitations can be assigned with the Cartesian quantum numbers a v(x), v(y), and v(z). The radial anisotropy of the cage environment causes the splitting of the translational fundamental into three excitations whose frequencies differ substantially; the z mode directed toward the ATOCF orifice has the lowest frequency and is the most anharmonic. All three translational modes exhibit negative anharmonicity. The j = 1 rotational level of H-2 is also split into a triplet, due to the angular anisotropy of the cage. The complete lifting of the degeneracies of the translational fundamental and the j = 1 triplet of the encapsulated H-2 molecule that emerges from the calculations is also observed in the INS spectra of H-2@ATOCF. The calculated magnitudes of both splittings, as well as the energies of the individual sublevels, rotational and translational, are in good agreement with the INS data.