We consider seven fundamental properties of cellular embeddings of graphs in compact surfaces, and show that each property can be associated with a point of the Fano plane $F$, in such a way that allowable combinations of properties correspond to projective subspaces of $F$. This Fano framework allows us to deduce a number of implications involving the seven properties, providing new results and unifying existing ones. For each property, we provide a correspondence between embeddings with that property and an associated structure for $4$-regular graphs, using the medial graph of the graph embedding. We apply this to characterize when a graph embedding has a twisted dual with one of the properties. For each allowable combination of properties, we show that a graph embedding with these properties exists. We investigate connections between the seven properties and three weaker `Eulerian' properties. Our proofs involve parity conditions on closed walks in an extended version of the `gem' (graph-encoded map) representation of a graph embedding.