[en] Scattering of x-rays enhanced by an electric dipole-electric quadrupole (E1-E2) event probe parity-odd atomic multipoles at the heart of many current theories of complex materials, e.g., Mott-Hubbard insulators and multiferroics. We achieve a complete decomposition of the E1-E2 scattering amplitude at spin-orbit split states in terms of atomic multipoles, albeit unfamiliar composite atomic operators built with electron position, spin and orbital variables. In consequence, one has clear insight as to the physical makeup of quantities that determine structure factors for Bragg diffraction enhanced by resonant ions that occupy sites devoid of inversion symmetry. (author)