hi dulltheater,
When you sit bhind a right handed kit, where the hihat is on your left and the bassdrum is on your right, a right handed player will cross his right hand to his left to play the hihat and his left hand under his right to play the snare drum.
But for a left handed player to sit behind a right handed kit, the usual way is to use his left hand to play the hihat and the right hand play the snare drum. This is what we call "Open Handed". The ride cymbal will also be moved to the left hand side of the kit, very close to the hi-hat.
This is not limited to just left handed palyers. Some right handed players opt to play this way too, e.g. Billy Cobham, Simon Phillips and Dom Famularo. This method creates a whole lot of rythmic possibilites compared to the "traditional" way of crossing over.
Cheers....