[Stickman]

Cross handed playing began in the early 1900's with the advent of trapsets used in the traps for small plays, small orchestra, and vaudeville trap performers.
Mostly it was the result of weak left hands, so the dominant hand crossed over to play the more articulate parts.
Some drummers naturally gravitate to open handed playing, although it is somewhat rare.
Open handed playing is slowly developing in to becoming the standard for proper technigue and freedom of expression within set performance.
With todays advancements in stands and gear it makes it a great deal easier to move to open handed methods of playing.
All that having been said, my reccomendation is to experiment with playing styles as much as possible.
Try reversing your setup just to give it a try. It can be a little confusing at first as your instintcs will have to be resisted a bit; but, with just a fews days of effort it can pay off and open a whole realm of possibilities.