Rotational tunneling of tetrahedral molecules is a sensitive probe of site symmetries and site multiplicities. However, the complexity of overlapping tunneling patterns can render a model free description of data impossible. Only on the basis of the recently determined low temperature crystal structure of phase III of methane the high resolution tunneling spectrum of 1.5% CH₄ in CD₄ could be quantitatively explained. The analysis is based on tunneling matrix elements and includes line positions and intensities. Barrier heights and disorder obtained with accuracies better than O.1meV will allow to refine pair interaction potentials. Level specific spin conversion times show up in the intensities of tunneling transitions.