A spanning tree connects every vertex of a graph without making a cycle. A classical theorem says that the number of spanning trees equals the size of another object, called the critical group. But can that group act on the trees in a way that respects every symmetry of the graph? This candidate gives an exact criterion, and a way to count and construct compatible actions. The stronger family result concerns any number of separate paths joining two terminals. For a simple graph with at least three such paths, a compatible action exists exactly when the path lengths are all different and at least one length is even. Repeated lengths create an obstruction because swapping equal paths fixes the wrong number of trees. Distinct lengths leave a reversal symmetry, whose answer is controlled by parity. This has a surprising consequence. Replacing every edge by two edges can change a negative graph into a positive one, even though its topology and abstract symmetry group stay the same. The paper also counts the possible torsor classes, showing that existence can be very far from uniqueness. The package contains the written proofs and exact checks, including a separate Laplacian calculation on one hundred and twenty-three graphs. Those finite checks do not prove the universal theorem. This is an unrefereed Evidence Press candidate. The basic tools are classical, and historical priority and external specialist validation remain unestablished. The full paper and evidence are linked. This is an AI-generated voice summary, not additional mathematical evidence.