We derive computable formulas for the structured backward errors of a complex number $λ$ when considered as an approximate eigenvalue of rational matrix polynomials that carry a symmetry structure. We consider symmetric, skew-symmetric, T-even, T-odd, Hermitian, skew-Hermitian, $*$-even, $*$-odd, and $*$-palindromic structures. Numerical experiments show that the backward errors with respect to structure-preserving and arbitrary perturbations are significantly different.