The crystal structures of type II ${\mathrm{N}}_2\mathrm{-,}$ ${\mathrm{O}}_2\mathrm{-,}$ and air-clathrate hydrates have been studied under their thermodynamic conditions of stability by neutron powder diffraction. The isothermal compressibilities were deduced from the refined lattice constants and found to be significantly different for ${\mathrm{N}}_2$ and ${\mathrm{O}}_2$ clathrate hydrates. The water host lattice structure rearranges as a function of pressure with a resulting change of the volume ratio of small and large cages. The cage fillings were established on an absolute scale for the first time. The large cages were found to be partly occupied by two molecules. The results are compared with predictions based on the statistical thermodynamic theory of van der Waals and Platteeuw. While the filling follows the predicted trend significant differences exist in detail.