So here's a nice little article on another site:
The cat stance represents a weight shift:
So - you're either shifting your weight to empty the leg to kick or you're shifting your weight because you are using it to disrupt the balance of your attacker. That can be in several ways:
1. You've grabbed onto something of theirs that you're using like a level to pull them off balance (jerking power)
2. you've set up a lock of some kind and you're dropping your weight to apply pressure to break/get pain compliance
3. You're doing both the above at the same time (personal favorite)
4. You're shifting yourself off the line of attack (by a very little bit or more depending on how much time you have and how much you move.
Now - here's where the cat stance turn comes in. Sometimes you want to jerk someone in a circular fashion instead of a linear one. This is one of the main principles of small circle ju-jitsu. You turn a smaller faster circle on the inside than they can do on the outside and so they lose.
So now you've got something that allows you to do four things at the same time if you're not just kicking:
use stepping and shifting to get off the line of attack, use the shifting your bodyweight to displace by jerking, use the shifting of your bodyweight to break/get pain compliance, use the twisting of the cat stance turn to double your jerking power by doing it in a circle.
The question remains - do you need to come to the twisted stance to get this done or can you do the same thing through the regular sanchin turn?