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We have found that the polarization dependence of Raman scattering in organic crystals at finite temperatures can only be described by a fourth-rank formalism. This generalization of the second-rank Raman tensor $\mathcal{R}$ stems from the effect of off-diagonal components in the crystal self-energy on the light scattering mechanism. We thus establish a novel manifestation of phonon-phonon interaction in inelastic light scattering, markedly separate from the better-known phonon lifetime.