Full wave solutions are derived for vertically and horizontally polarized waves diffusely scattered across an interface that is two-dimensionally rough separating two different propagating media. Since the normal to the rough surface is not restricted to the reference plane of incidence, the waves are depolarized upon scattering; and the single scattered radiation fields are expressed as integrals of a surface element transmission scattering matrix that also accounts for coupling between the vertically and horizontally polarized waves. The integrations are over the rough surface area as well as the complete two-dimensional wave spectra of the radiation fields. The full wave solutions satisfy the duality and reciprocity relationships in electromagnetic theory, and the surface element scattering matrix is invariant to coordinate transformations. It is shown that in the high-frequency limit the full wave solutions reduce to the physical optics solutions, while in the low-frequency limit (for small mean square heights and slopes) the full wave solutions reduce to Rice's (1951) small perturbation solutions. Thus, the full wave solution accounts for specular point scattering as well as diffuse, Bragg-type scattering in a unified, self-consistent manner. It is therefore not necessary to use hybrid, perturbation and physical optics approaches (based on two-scale models of composite surfaces with large and small roughness scales) to determine the like- and cross-polarized fields scattered across the rough surface.