An approach is presented for the inversion of simulated and experimental in-plane, co-polarized light scattering data in p and s polarization to obtain the normalized surface-height autocorrelation function and the rms-roughness of a two-dimensional randomly rough metal surface. The approach is based on an expression, obtained on the basis of second-order phase perturbation theory, for the contribution to the mean differential reflection coefficient from the light scattered diffusely by the rough surface. The inversion scheme is validated by using several sets of computer generated scattering data for rough silver surfaces defined by Gaussian surface height correlation functions. The reconstructions obtained by this approach are found to be rather accurate for weakly rough surfaces illuminated by p- and s-polarized incident light; this is also true in cases where the contributions to the input data from multiple scattering of surface plasmon polaritons is not insignificant. Finally, the inversion scheme is applied to experimental scattering data obtained for characterized two-dimensional randomly rough gold surfaces, and the results obtained in this way, compare favorably to what is obtained by directly analyzing the surface morphology. Such results testify to the attractiveness of the computationally efficient inversion scheme that we propose.