Let A = P + K be an n × n complex matrix with P = 1 2 (A − HAH) and K = 1 2 (A + HAH), H being a unitary involution. Having characterised all unitary involutions, we investigate the spectral structure of P and K and, in particular, characterise the eigenvalues of K as zeros of a rational function and prove that, for normal A, σ(K) resides in the convex hull of σ(A). We also demonstrate that this need not be true when A is not normal.