My question is really about the gravitomagnetic frame-dragging and the Lense Thirring effect. My question is not whether the frame dragging effect exists but rather it's manifestation in affecting precession. From Wikipedia:
Josef Lense and Hans Thirring ... predicted that the rotation of a massive object would distort the spacetime metric, making the orbit of a nearby test particle precess.
If I understand it right we can extrapolate the effect to frame dragging in general. So let's say we have a large rotating ball floating on the surface of a swimming pool, dragging the water around it and there is a smaller ball rotating nearby. To me the scene looks perfectly symmetrical and I can not see where the precession is coming from.