Given a matrix belonging to some class of structured matrices, we consider the question of comparing the sensitivity of its eigenvalues under two different kinds of perturbations, either unstructured (i.e., arbitrary) or structured (i.e., those belonging to the same class of matrices as the unperturbed one). In a previous paper (Kressner et al., 2009 [13]), the authors compared the structured and unstructured condition numbers of (possibly multiple) eigenvalues for several different matrix and pencil structures. Only one case was left out of the analysis, namely the one where the asymptotic order of perturbed eigenvalues under structured perturbations is different from the asymptotic order under unstructured ones. This is precisely the case we consider in the present paper: given a matrix which is skew-adjoint with respect to a symmetric scalar product and has a zero eigenvalue with a certain Jordan structure, first order expansions are obtained for the perturbed eigenvalues under structured perturbation, as well as bounds on the structured condition number. Similar results are obtained for structured perturbations of symmetric/skew-symmetric and palindromic matrix pencils. (C) 2013 Elsevier Inc. All rights reserved.