  The correlation between local structure and the propensity for structural
rearrangements has been widely investigated in glass forming liquids and
glasses. In this paper we use the excess two-body entropy $S_2$ and
tetrahedrality $\n_{tet}$ as the per-particle local structural order parameters
to explore such correlations in a three-dimensional model glass subjected to
cyclic shear deformation. We first show that for both liquid configurations and
the corresponding inherent structures, local ordering increases upon lowering
temperature, signaled by a decrease in the two-body entropy and an increase in
tetrahedrality. When the inherent structures, or glasses, are periodically
sheared athermally, they eventually reach absorbing states for small shear
amplitudes, which do not change from one cycle to the next. Large strain
amplitudes result in the formation of shear bands, within which particle motion
is diffusive. We show that in the steady state, there is a clear difference in
the local structural environment of particles that will be part of plastic
rearrangements during the next shear cycle and that of particles that are
immobile. In particular, particles with higher $S_2$ and lower $n_{tet}$ are
more likely to go through rearrangements irrespective of the average energies
of the configurations and strain amplitude. For high shear, we find very
distinctive local order outside the mobile shear band region, where almost
$30\%$ of the particles are involved in icosahedral clusters, contrasting
strongly with the fraction of $<5\%$ found inside the shear band.
