Yes it is possible to fit it back to front , if you look at the diaphram side on, you will see it is slightly dome shaped. The dome points towards the front of the ball valve (the bit where the arm is attached). As the arm lowers then the diaphram springs forward clearing the inlet hole and allowing water to flow in.
... if the cistern takes ages to fill then my first port of call would be to get in the loft with a torch and have a look inside the tank. Make sure that the outlet at the bottom of the tank is not blocked with ***. Take the opportunity to drain it down and clear it out if it if it is. Over the years I have found rats, mice, and even pigeons festering in the tank. All of these breakdown into fragments as they decompose and eventually find their way into the pipe that feeds your cistern. So any tiny fragment will block the hole that you have in your inlet valve. The most difficult one I have ever done had the feathers from a pigeon blocking the pipe, it took a good few blasts with mains pressure from a hose to clear it. Make sure you remove the ball valve in the cistern if you decide to go this far. It will allow the pipe to clear quicly if there is any debris in it.
One more point is to check you dont have a flow restrictor fitted inside the inlet pipe near the cistern.
Good luck !