  The mechanical response of glasses remains challenging to understand. Recent
results indicate that the oscillatory rheology of soft glasses is accompanied
by a sharp non-equilibrium transition in the microscopic dynamics. Here, we use
simultaneous x-ray scattering and rheology to investigate the reversibility and
hysteresis of the sharp sharp symmetry change from anisotropic solid to
isotropic liquid dynamics observed in the oscillatory shear of colloidal
glasses [D. V. Denisov, M. T. Dang, B. Struth, A. Zaccone, and P. Schall, Sci.
Rep. 5, 14359 (2015)]. We use strain sweeps with increasing and decreasing
strain amplitude to show that, in analogy to equilibrium transitions, this
sharp symmetry change is reversible and exhibits systematic frequency-dependent
hysteresis. Using the non-affine response formalism of amorphous solids, we
show that these hysteresis effects arise from frequency-dependent non-affine
structural cage rearrangements at large strain. These results consolidate the
first-order like nature of the oscillatory shear transition and quantify
related hysteresis effects both via measurements and theoretical modelling.
