PID controller - should terms be completely independant?
Yes, it's to prevent overshoot. The PID controller output will saturate so driving the integral term to the opposite extreme can prevent a large overshoot. 

There are many approaches to this problem.

The basic issue is that the integral term continues to integrate even with the output saturated, but even in an ideal situation with a PID controller there is overshoot which may be undesirable. This is also called "reset windup".