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We isolate and study the transformation of the intrinsic spin of photons and Dirac particles as they propagate along geodesics in the outer geometry of a Kerr black hole. We show how observers located in the outer geometry of Kerr black holes may exploit the symmetries of the geometry to set up reference frames using purely geometric, locally-available information. We investigate the purely geometrically-induced precession of the spin of photons and Dirac particles.