Second-order [ O(k(2)), k = omega/c] nondipole effects in soft-x-ray photoemission are demonstrated via an experimental and a theoretical study of angular distributions of neon valence photoelectrons in the 100-1200 eV photon-energy range. A newly derived theoretical expression for nondipolar angular distributions characterizes the second-order effects using four new parameters with primary contributions from pure-quadrupole and octupole-dipole interference terms. Independent-particle calculations of these parameters account for a significant portion of the existing discrepancy between experiment and theory for Ne 2p first-order nondipole parameters.