On the basis of the recently determined low temperature crystal structure of phase III of methane new high resolution tunneling spectra on 1.5% CH4 dissolved in CD4 are successfully described by two tunneling systems at sites with two-fold and mirror symmetry, respectively. The analysis is based on tunneling matrix elements. The octopole moments of the CH4 impurities are found to agree with those of the CD4 host molecules. T levels of molecules at m sites show different spin conversion times. Spectra of pure CD4 III are reanalyzed and consistently explained assuming the presence of a small contamination of CD3H.