In this paper, we consider symmetric binary programs that contain set packing, partitioning, or covering inequalities. To handle symmetries as well as set packing, partitioning, or covering constraints simultaneously, we introduce constrained symresacks which are the convex hulls of all binary points that are lexicographically not smaller than their image w.r.t. a coordinate permutation and which fulfill packing, partitioning, or covering constraints. We show that linear optimization problems over constrained symresacks can be solved in cubic time. Furthermore, we derive complete linear descriptions of constrained symresacks for particular classes of symmetries. These inequalities can then be used as strong symmetry handling cutting planes in a branch-and-bound procedure. Numerical experiments show that we can benefit from incorporating set packing, partitioning, or covering constraints into symmetry handling inequalities.