  We derive computable formulas for the structured backward errors of a complex
number $\lambda$ 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.
