Moving media break time-reversal symmetry and exhibit intriguing optical nonreciprocity. This nonreciprocity is usually weak due to the much lower moving speed of media relative to the speed of light. We demonstrate that strong optical nonreciprocity can emerge in a two-dimensional photonic crystal composed of spinning dielectric cylinders. The photonic crystal supports two types of chiral modes at the Brillouin zone center: hybridized multipole modes and symmetry-protected bound states in the continuum (BICs), both of which carry intrinsic spin angular momentum. For finite wavevectors near the zone center, the BICs transform into quasi-bound states in the continuum (QBICs). Under oblique incidence of circularly polarized plane waves, the photonic crystal exhibits nonreciprocal transmission and absorption that are significantly enhanced at the frequencies of these hybridized multipole modes and QBICs. Furthermore, the high quality factors of the QBICs enable sharp transitions in nonreciprocity. Our work uncovers strong chiral light-matter interactions in periodic moving structures, with potential applications in nonreciprocal light manipulation. The mechanism may also be generalized to other classical wave systems, such as phononic crystals.