[Alan Anderson]

If the problem were only the wheels, it would drive forward and strafe okay but fail to rotate well.

The trouble with mecanum drive is that there are so many places to get things reversed. A miswired motors or wrongly-mounted wheels or swapped PWMs or mismatched program definitions will cause very confusing results. The only reasonably way to get it to work properly is to make sure to keep the hardware and software in perfect synchronization from the beginning. Once it's messed up, you have to focus on one motor at a time and verify that it moves when it should and in the direction it should. Do that with each motor in turn, correcting the wiring and/or programming as necessary.

And recognize that if you don't use some form of closed-loop control, weight balance will become extremely important.