I’m not sure what you mean by “deep into the backhand.” For clarity I’ll describe side slice as moving into your body or away from your body.
A rightie bh slice is will tend to move away from your body if you’re taking it as a fh (rightie). The lower your opponent hits the ball the more side slice it will tend to have. If the ball is high, say above the waist or so, it won’t have very much side slice, if any. I don’t see how a right handed player could produce a backhand slice that has a side component in the opposite direction. It’s a function of the arm pivoting at the shoulder, and with a slice the racquet will start away from the body and move in an arc down and towards the body. You’re never going to start a slice over your head and move the racquet to the outside, so I don’t see how you could get side motion on the ball that curves into a rightie fh.
A topspin slice fh, like the one you’re describing from Cheetah, can be more varied. In this case you can generate a swing that moves away from the body or towards the body depending on the height of the ball, thereby generating a side slice that can move in either direction. If the ball is low a player can basically hit around the outside of the ball (LeeD talks about this all the time). This would create a ball like you describe that would move away from your body on the fh side. I think it’s a bit easier to do this with a more Western grip like the one Cheetah uses. On high balls you can get the opposite, a ball that moves into your body on the fh side. Good players like Cheetah are also capable of varying the amount of side spin on the ball, though the overall direct of the side spin is mostly a function of height. The high balls with side spin are generally the ones that I’ve noticed that can kick differently than how they’ve been moving in the air. It can be really difficult to read exactly how the spin is going to make the ball move and bounce, and all of this makes them harder to handle. You have to watch the swing and watch the ball all the way to your racquet, and be balanced enough in your set-up to adjust to last second surprises. The shot is supposed to cause you problems, so don’t feel bad that it does.
Does this help?