Two things to deal with here.
You don't really need a reversible motor for cutting left hand threads. You might for cutting metric threads on a machine with an imperial lead-screw. For LH threads just set your tumbler to feed the saddle towards the tail-stock.
From what I can make of the diagram in the hyperlink K1 and K2 are the run windings, D1 and D2 are the start windings. The Santon/Dewhurst (or similar) switch would be wired to reverse the phase relationship of the two windings to change the direction of motor rotation.
Come back to us if you need any more help.