Cutting planes have been an important factor in the impressive progress made by integer programming (IP) solvers in the past two decades. However, cutting planes have had little impact on improving performance for symmetric IPs. Rather, the main breakthroughs for solving symmetric IPs have been achieved by cleverly exploiting symmetry in the enumeration phase of branch and bound. In this work, we introduce a hierarchy of cutting planes that arise from a reinterpretation of symmetry-exploiting branching methods. There are too many inequalities in the hierarchy to be used efficiently in a direct manner. However, the lowest levels of this cutting-plane hierarchy can be implicitly exploited by enhancing the conflict graph of the integer programming instance and by generating inequalities such as clique cuts valid for the stable-set relaxation of the instance. We provide computational evidence that the resulting symmetry-powered clique cuts can improve state-of-the-art symmetry exploiting methods. The inequalities are then employed in a two-phase approach with high-throughput computations to solve heretofore unsolved symmetric integer programs arising from covering designs, establishing for the first time the covering radii of two binary-ternary codes.