I confirmed with Hiro my suspicion that the situation described in the quote probably can't really be what's going on. As in the case of a regular fabry-perot cavity, the mirrors themselves define the mode and they must move for any rotation or translation of the mode to take place. So, pointing drift in the input beam can't cause a differential path length between the CCW and CW modes, though it would cause us to see AM at the output (like we do) as the coupling would go up and down.
I guess it's possible that the mirrors themselves can be twisting and causing the mode to wander, but I think geometry dictates that the effect would be the same for both directions.
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I can't say that I can think of any obvious coupling mechanism here. I think it's feasible that pointing drift of the input beam is causing the spatial eigenmodes of the cavity to wander differentially (i.e., the points at which the supported mode in one direction touches the mirrors all move horizontally across the mirrors a bit, making the whole square 'rotate'---this is dependent on the input beam and can thus happen independently in each direction). In this case, the cavity length would differ in the two directions, resulting in a gyro signal.
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