Symmetry-based pruning is a powerful method for reducing the search effort in finitedomain planning. This method is based on exploiting an automorphism group connected to the ground description of the planning task - these automorphisms are known as structural symmetries. In particular, we are interested in the STRUCTSYM problem where the generators of this group are to be computed. It has been observed in practice that the STRUCTSYM problem is surprisingly easy to solve. We explain this phenomenon by showing that STRUCTSYM is GI-complete, i.e., the graph isomorphism problem is polynomial-time equivalent to it and, consequently, solvable in quasi-polynomial time. This implies that it is solvable substantially faster than most computationally hard problems encountered in AI. We accompany this result by identifying natural restrictions of the planning task and its causal graph that ensure that STRUCTSYM can be solved in polynomial time. Given that the STRUCTSYM problem is GI-complete and thus solvable quite efficiently, it is interesting to analyse if other symmetries (than those that are encompassed by the STRUCTSYM problem) can be computed and/or analysed efficiently, too. To this end, we present a highly negative result: checking whether there exists an automorphism of the state transition graph that maps one state s into another state t is a PSPACE-hard problem and, consequently, at least as hard as the planning problem itself.