I'm looking for as many characterizations of planar graphs, preferably those that are more `structural'. Wagner's and Kuratowski's results get close, but the characterizations of Whitney and Maclane is more what I'm looking for. That is, graphs are related to some other objects (matroids and binary vector spaces, respectively), and are then described with respect to those. There's also the lesser known Schnyder's theorem (https://en.wikipedia.org/wiki/Schnyder%27s_theorem) that fits the bill.
I should emphasize that I am not interested in characterizations related to drawings of the graph, such as Hanani–Tutte.