Orienting a Pyrene diagram

Another example from Symmmetry in Chemistry, of drawing pyrene:

the standard (IUPAC?) orientation on the left doesn't show the symmetries of the molecule as well as the rotated version on the right. The letters indicate equivalent atom positions - oh, and aromatic indicators are missing :)

It seems like there should be a way to discover symmetry axis from the graph - without coordinates. In a similar way that some ring perception algorithms work by reducing the graph:

However, it is not obvious that this would be better than laying out the structure, then using the 2D coordinates to determine symmetries. Also not clear to me is how to choose which vertices to merge, and which to duplicate. On the left hand side of the image above, the fragments have duplicate (a), (d), and (e) vertices, but (e) has also been merged before being duplicated.