I had a lefty, once. I liked to think of it as the Worlds Ugliest Fixie (and that's saying something!) Mid-70s Univega Viva Sport with faded and peeling-off woodgrain vinyl shelf paper wrapped around the tubes, handlebars and seatpost, saddle slathered with about an inch thickness of black duct tape, dried out knobby tires, and enough thread lock to keep the pedals in place. I think I remember the guy I bought it from ($50) telling me to keep an eye on the pedals if I was actually going to ride it.
After I put together a proper fixie in my size, I built the old one up as a decent commuter bike for my sister, replacing her badly bent Motobecane.