A spectrahedron is the solution set to a linear matrix inequality. Consider a spectrahedral cone K in n-space which is symmetric with respect to permuting the coordinates. According to an observation by Bauschke, Güler, Lewis and Sendov the set S(K) of all symmetrich nxn matrices, whose spectrum is in K, is a hyperbolicity cone. We give a representation theoretic sufficient condition on K for S(K) being a spectrahedral cone. Applying this to Brändén's spectrahedral representation of elementary symmetric polynomials yields a spectrahedral representation of all derivative relaxations of the cone of positive semidefinite matrices.