Pruning techniques based on partial order reduction and symmetry elimination have recently found increasing attention for optimal planning. Although these techniques appear to be rather different, they base their pruning decisions on similar ideas from a high level perspective. In this paper, we propose safe integrations of partial order reduction and symmetry elimination for cost-optimal classical planning. We show that previously proposed symmetry-based search algorithms can safely be applied with strong stubborn sets. In addition, we derive the notion of symmetrical strong stubborn sets as a more tightly integrated concept. Our experiments show the potential of our approaches.