Scattering of scalar and vector waves from a randomly rough interface between media, in which several types of waves (modes) exist due to the time and spatial dispersion, has been studied in the Kirchhoff approximation. As a wave of a certain type is reflected from the interface, it transforms into other modes not only in the diffusive fields but in the coherent components as well. We have calculated the mean (coherent) field and the angular diagram of the diffusively scattered intensity. It is shown that the coherent components of the waves generated on reflection (cross-modes) propagate in directions that are different from the specular one. The dispersion gives rise to the frequency dependence of the scattering diagram even in the geometric-optics approximation.