We report rigorous quantum calculations of the translation-rotation (T-R) eigenstates of the H2 molecule in C60. The resulting level structure can be explained in terms of a few dominant features. These include the coupling between the orbital and the rotational angular momenta of H2 to give the total angular momentum lambda, and the splitting of the sevenfold degeneracy of T-R levels with lambda=3 by the nonsphericity of C60, according to the rules of the icosahedral I h group.