We report fully coupled quantum six-dimensional (6D) calculations of the translation-rotation (T-R) energy levels of CH4 molecule inside the small dodecahedral (512) and large tetracaidecahedral (5 1262) cages of the structure I clathrate hydrate. The quantum dynamics of the three translational and three rotational degrees of freedom of CH4 are treated rigorously, while the guest molecule and the host cavities are taken to be rigid. The matrix of the full 6D T-R Hamiltonian is diagonalized in the product basis of contracted translational and angular basis functions, generated by solving two reduced-dimension (3D) eigenvalue problems. A pairwise additive CH4-cage 6D potential energy surface (PES) is employed, constructed using the anisotropic CH 4H2O pair potential which was utilized previously in the molecular dynamics simulations of methane hydrate. Our calculations elucidate the key features of the T-R energy level structure of the nanoconfined CH 4. The rotational levels of methane exhibit an elaborate pattern of splittings caused by the angular anisotropy of the environment; the splitting patterns are identical for both types of cages. Translationally excited T-R states in the small cage are assigned in terms of the quantum numbers n and l of the 3D isotropic harmonic oscillator and those in the large cage using the Cartesian quantum numbers. Extensive comparison is made with the data from the inelastic neutron scattering studies of methane hydrate, allowing an assessment of the accuracy of the 6D PES employed.