Intense peaks are observed in angular intensity distributions for HD scattering from the Cu(001) surface. These can be ascribed to coherent diffraction with translational energy transfers of DeltaE=0, -11.0, or 33.11 meV to molecular rotational energy. Time-of-flight spectra, at other scattering angles, display inelastic peaks that are assigned to phonon creation or annihilation processes with either the HD rotationally elastic or inelastic transitions. The HD phonon dispersion curves suggest a strong HD coupling both with surface Rayleigh modes and with bulk phonon modes of the metal surface. Comparisons of both elastic and inelastic scattering intensities are also made with those reported for the Ni(001) surface. To explain an apparent anomalous diffraction peak intensity ratio for Ni, a preferential coupling for the HD J=0 rotational state into a predissociation channel above the Ni surface is proposed.