I have a '00 Craftsman 21HP I/C twin. She runs like a top-unless.
When mowing on a side hill after about a minute it starts to sputter and backfire. If I turn into the hill or down it goes away. This only happens with the left side on the downhill?
This has been happening for quite a while now. The oil is up-so it is not low oil sensor. It happens periodically? Seemingly more so on hot days than cool or after it has been run quite a bit.
I have secured the safety switch in the seat to see if that is the problem but thats not it. It feels and sounds as if that switch was activated by taking your butt off the seat while underway.
The float in the fuel bowl is in the center so that should not effect the right or left side lean? What could it be?

well, a severe hill may cause the carb not to act right. though if its not much of a hill, it may need a tiny bit of a carb adjustment to run in the heat, though all the newer ones, you can't hardly do much to em.

This is just a guess, but it sounds like your float is hanging up and drowning the engine with fuel. This would explain why it's worse when the engine is hot because the air is less dense and it can't burn all the fuel.