Mine all turn in the normal clockwise direction looking down the tap into the hole, like a drill.
I've not used mine on a left hand tap, but maybe the mechanism is adjustable to set it to run in reverse for left handed taps.
Not sure what make mine are, will have to check, one I think is an old archer tapping head.
Does yours let you pull the tap down so you can twist it to stay in that down position?
I think I may have the instruction card in the boxes for a couple of them will check tomorrow