Non-Hamiltonian spectral conditions for the class of symmetric multivariable strictly positive real and strictly negative imaginary systems are derived. They represent generalizations of known ones for strict positive realness to the cases with singular feedthrough matrix, and are novel in the context of strict negative imaginariness. Moreover, we propose a concept of strong negative imaginariness and establish its links to strict positive realness of symmetric systems. The proposed spectral conditions are useful in the corresponding assessment and enforcement procedures, as well as in quadratic stability analysis of uncertain and switched systems.