Symmetry handling inequalities (SHIs) are a popular tool to handle symmetries in integer programming. Despite their successful application in practice, only little is known about the interaction of SHIs with optimization problems. In this article, we focus on SST cuts, an attractive class of SHIs, and investigate their computational and polyhedral consequences for optimization problems. After showing that they do not increase the computational complexity of solving optimization problems, we focus on the stable set problem for which we derive presolving techniques based on SST cuts. Moreover, we derive strengthened versions of SST cuts and identify cases in which adding these inequalities to the stable set polytope maintains integrality. Preliminary computational experiments show that our techniques have a high potential to reduce both the size of stable set problems and the time to solve them.