  The rheology of non-colloidal 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 ($N_2$).
Specifically, we find a reduced viscosity, an enhanced intracycle shear
thinning, the onset of a finite $N_2$ and its frequency doubling, all near a
critical strain amplitude $\gamma_c$ that scales with the volume fraction
$\phi$ as $\gamma_c \sim \phi^{-2}$. 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 towards
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.
