The pure rotational spectrum in the far-infrared and its absolute intensity in the vibrational ground state of CHD3 and CH3D, and the integrated band strength of the N=5 CH-stretching overtone of CHD, in the near infrared to visible were measured by high-resolution interferometric Fourier transform techniques. The far-infrared data result in permanent electric dipole moments (mu(0)(z)=(5.69+/-0.14)x10(-3) D for CHD3, mu(0)(z)=(5.57+/-0.10)x10(-3) D for CH3D), consistent with previous experimental data. The integrated N=5 overtone cross section is found to be (0.828+/-0.068) fm(2). The overtone data are used, together with previous data, to derive a new, nine-dimensional, isotopically invariant dipole moment function for CH4 within the chromophore model for the CH chromophore in CHD3. With this function, the experimental data can be reproduced to an averaged factor of 1.2, in the best case. In the vibrational ground state, a nine-dimensional calculation of expectation values on a new, fully anharmonic potential surface was performed using the solution of the rovibrational Schrodinger equation by diffusion quantum Monte Carlo methods. The results for the rotational constants of several isotopomers, which include significant contributions from rovibrational interactions, indicate that the equilibrium CH bond length of methane is r(e)=108.6 pm. The calculated value for the vibrationally averaged permanent dipole moment from these nine-dimensional vibrational quantum calculations, using the dipole moment function consistent with the analysis of the overtone bands, is mu(0)(z) =(6.8+/-0.5)X10(-3) D for CHD3 (with positive z coordinate for the H atom) and mu(0)(z)=(6.8+/-0.5)X10(-3) D for CH3D (with positive z coordinate for the D atom) in essential agreement with the far-infrared rotational intensities. The sign could be determined unambiguously by comparison with ab initio data. We predict the permanent dipole moment of several further methane isotopomers. The polarity of the CH bond in methane is C--H+, within our simple bond dipole model, but is discussed to be a model dependent (not purely experimental) quantity.