We report quantum five-dimensional (5D) calculations of the energy levels and wave functions of the hydrogen molecule, para-H-2 and ortho-H-2, confined inside the small dodecahedral (H2O)(20) cage of the sII clathrate hydrate. All three translational and the two rotational degrees of freedom of H-2 are included explicitly, as fully coupled, while the cage is treated as rigid. The 5D potential energy surface (PES) of the H-2-cage system is pairwise additive, based on the high-quality ab initio 5D (rigid monomer) PES for the H-2-H2O complex. The bound state calculations involve no dynamical approximations and provide an accurate picture of the quantum 5D translation-rotation dynamics of H-2 inside the cage. The energy levels are assigned with translational (Cartesian) and rotational quantum numbers, based on calculated root-mean-square displacements and probability density plots. The translational modes exhibit negative anharmonicity. It is found that j is a good rotational quantum number, while the threefold degeneracy of the j = 1 level is lifted completely. There is considerable translation- rotation coupling, particularly for excited translational states.