Yet another Southpaw here. I got forced to do the right handed writing thing. I notice I switch depending on the surface; when using paper and pencil on a flat horizontal surface I'm using my right hand, but when I'm writing on a vertical surface, like a chalk board (yes those are still in classrooms!) I use my left hand. To top if off both examples look completely different.
Strange.... I ended up playing one of the few brass instruments in which you use both hands.