Pfaffian graphs, T-joins and crossing numbers

We characterize Pfaffian graphs in terms of their drawings in the plane. We generalize the techniques used in the proof of this characterization, and prove a theorem about the numbers of crossings in T-joins in different drawings of a fixed graph. As a corollary we give a new proof of a, theorem of Kleitman on the parity of crossings in drawings of K2j+1 and K2j+1.2k+1.