Symmetry properties of the Zwanzig-Fano relaxation matrix are studied. Its invariance under rotations and inversion is proven for isotropic gases, to all orders in the gas density. Each multipole radiation operator is confined to a distinct invariant subspace in the Liouville space of operators. These invariant subspaces form the basis for the reduction of the relaxation matrix; therefore, the various multipole spectra are broadened independently. Properties of the relaxation matrix under Liouville conjugation are studied, and expressions are given relating matrix elements in which Liouville-conjugate pairs of vectors are involved.