Detecting rotational symmetries of spatial structures
Here is one approach  to identify all rotational symmetries of the graph:

Pick any three vertices, $v_1,\dots,v_3$.  Loop over all possible combinations of three vertices $w_1,\dots,w_3$.  For each such combination, find the unique rotation that maps $v_1 \mapsto w_1,\dots,v_3 \mapsto w_3$, then check whether this is a rotational symmetry of the entire graph (by mapping the entire graph and seeing whether the mapped graph coincides with the original graph).

The worst-case running time will be something like $O(n^4)$, where $n$ is the number of vertices in the graph.  In practice I expect it to be typically more like $O(n^3)$, as most candidate rotations can be immediately rejected by testing a few vertices.

To identify the unique rotation that maps $v_1 \mapsto w_1,\dots,v_3 \mapsto w_3$, given $v_1,\dots,v_3,w_1,\dots,w_3$, see https://math.stackexchange.com/q/3341635/14578.