This paper deals with the study of the following class of hemivariational inequalities: find \(u\in H^1(\mathbb R^N)\) such that \[ \int_{\mathbb R^N}(\nabla u\nabla w+uw)dx+\int_{\mathbb R^N}F^0_x(x,u(x);-w(x))dx\geq 0, \] for all \(w\in H^1(\mathbb R^N)\), where \(F^0\) stands for the generalized directional derivative in the sense of Clarke. Under some natural symmetry assumptions, the main results of the paper establish the existence of infinitely many radial solutions, resp. of infinitely many non-radial solutions. The main tool used in the proofs is the principle of symmetric criticality for a locally Lipschitz functional which is invariant under a group action. The reviewer believes that this method can be extended for the qualitative analysis of large classes of nonsmooth nonlinear problems.