  We consider a ferromagnetic spin-1 Bose-Einstein condensate (BEC)
dispersively coupled to a unidirectional ring cavity. We show that the ability
of a cavity to modify, in a highly nonlinear fashion, matter-wave phase shifts
adds a new dimension to the study of spinor condensates both within and beyond
the single-mode approximation. In addition to demonstrating strong matter-wave
bistability as in our earlier publication [L. Zhou et al., Phys. Rev. Lett.
103, 160403 (2009)], we show that the interplay between atomic and cavity
fields can greatly enrich both the physics of critical slowing down in spin
mixing dynamics and the physics of spin-domain formation in spinor condensates.
