The authors show that the only real symmetric matrices whose spectrum is invariant modulo sign changes after either row or column reversal are the centrosymmetric matrices; moreover, they prove that the class of real symmetric centrosymmetric matrices can be completely characterized by this property. The paper also shows that the only real symmetric matrices whose spectrum changes by multiplication by \(i\) after either row or column reversal are the skew-centrosymmetric matrices and that the class of real symmetric skew-centrosymmetric matrices can be completely characterized by this property of their eigenvalues. Both of these spectral characterizations are proved as special cases of results for generalized centrosymmetric \(K\)-matrices and generalized skew-centrosymmetric \(K\)-matrices.