To motivate the question, Andy Strominger recently put out a paper on calculating the Sagnac shift of counterrotating beams due to the angular momentum flux of a passing gravitational wave. See here: http://arxiv.org/abs/1502.06120. But consider now two nearby freely falling gyroscopes initially at rest in some background inertial frame and a weak gravitational wave passing by them. Is there a way to compute the differential precession of the gyroscopes due to the angular momentum of the gravitational wave, say by looking at the magnetic part of the Riemann tensor and/or the twist of a congruence of freely falling gyroscopes? Has such a calculation been done in the literature? To clarify, I don't mean the precession of a single gyroscope relative to the axes of the TT-gauge coordinates. Furthermore, would there also be a memory effect after the gravitational wave has gone to future null infinity, say in terms of a permanent relative shift of the gyroscopes' axes?