Helicopters do counter this with what is known as "flap" with complex rotor systems (fully articulated and semi rigid). However, the pilot's presence etc causes imbalances in the center of mass. This is meant for simulation. It's the pilot's duty to combat this.
The huey, for example, leans right and forward. That's why pilots use trim. MD Hughes 500 (littlebirds) lean different ways depending on their model. I've flown them.