The endohedral fullerene CH4@C84 has been studied using density functional theory (DFT) and second-order Møller-Plesset perturbation theory (MP2). In addition to the structure with a CH bond of CH4 in a tetrahedral pocket conformation, we find an alternative minimum, very close in energy (6.3-9.5 kJ/mol higher according to the level of theory), with the methane inverted, which we call the antipocket conformation. Computed IR spectra are reported for CH4@C84 and also for the reference system CH4@C60. The calculated vibrational levels, in a harmonic approximation, reveal close-lying translational, librational, and shell-vibrational modes. The results are also presented for the isoelectronic species NH@C60.