the working in reverse has to do with the brake bands being worn unevenly from being mostly used going forward.
to put it very simply - initially The joystick sends enough hydraulic pressure to dis-engage the clutches for slight steering, then if the joystick is pushed farther enough hydraulic pressure is sent to engage the brakes for full lock turning. This is why the linkage under the joystick has to be completely without slop and worn parts. Again, this is a very simplistic explanation of the steering, but if the linkage under the joystick is worn or out of adjustment then the hydraulic pressure sent to clutches and brakes isn't what it is supposed to be.
After I've done my best to adjust all joystick linkages and the brake adjustment in the back of the tractor I've decided my brakes are nearly worn out, most on the left. Clutch and brake replacement requires removing the "cab" and everything above them to access. It's above my ability to do by myself and I just play with my dozer on the farm so I live with it.
I hope this helps