The rheology of noncolloidal suspensions under cyclic shear is studied numerically. The main findings are a strain amplitude (gamma 0) dependent response in the shear stress and second normal stress difference (N2). Specifically, we find a reduced viscosity, an enhanced intracycle shear thinning, the onset of a finite N2, and its frequency doubling, all near a critical strain amplitude gamma c that scales with the volume fraction 0 as gamma c similar to 0-2. These rheological changes also signify a reversible-irreversible transition (RIT), dividing stroboscopic particle dynamics into a reversible absorbing phase (for gamma 0 < gamma c) and a persistently diffusing phase (for gamma 0 > gamma c). We explain the results based on two flow-induced mechanisms and elucidate their connection in the context of RIT through the underlying microstructure, which tends toward hyperuniformity near gamma 0 = gamma c. Overall, we expect this correspondence between rheology and emergent dynamics to hold in a wide range of settings where structural organizations are dominated by volume exclusions.