Well, if you have a big enough further-than-geosync counter-balance (say, much more than needed for the static part of the tether), the whole thing would work a lot like a pendulum. In other words, it would be in self-equilibrium due to constant, unidirectional tension.

My comment on stabilizing agents was more to stabilize spin around the axis of the tether itself, since that is only held still by the rotational stiffness itself. Given it is damn long, any realistic stiffness would still result in massive rotations up top, perhaps making it completely unfeasible for even having human beings inside it, let alone dock ships.