What the electromagnetic force in the motor does is to set up a 'push' between the rotor (attached to the platter) and the stator (attached to the boat, or the earth) . The push acts on both the rotor and the stator. If the stator is attached to something really massive (strictly something with a large moment of inertia, but that often involves a large mass) then it hardly moves. The rotor does all the moving and the platter goes round while the boat or earth stays almost still. The way that a force results in opposing motions is more familar to us if we think of motions in a straight line rather than rotary motions, but the principles are the same. If you've ever been in one boat and pushed against another boat you'll know that when you start pushing both boats move. If one is heavy and one is light then the heavy one doesn't move much and the light one moves a lot. And they move in opposite directions (away from one another). The same's true of rotation.
You're right about the twisting effect though. There really is nothing to worry about.