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\section{Supplementary Movie Caption:}

A movie of Brownian Dynamics simulation on a glassy hard sphere system under
oscillatory shear is shown, with stresses and structures depicted
simultaneously within the oscillation period. The movie is best viewed in a
loop.

Stresses are normalized by $K_{B}T/R^{3}$ and shown versus strain in a
Lissajous representation, together with the corresponding $g(r)-g_{rest}(r)$
in the velocity-gradient (xy) plane. A system with $\phi =0.60$ at four
different oscillation frequencies and strains is shown, corresponding to
non-linear regimes discussed in the main text.

Top figures (a, b) depict animations at intermediate strains with $\gamma
_{0}=0.3$, while bottom (c, d) at $\gamma _{0}=1.0$. On the left (a, c) we
show lower frequencies ($Pe_{\omega }=0.1$) while on the right (b, d) higher
ones ($Pe_{\omega }=100$). The movie supplements the information conveyed by
fig. 2 in the main text and expands the structural results to include
additional frequencies and strains as discussed in the text for experimental
stresses (fig. 1) and displacements from BD (fig. 4). One important finding
is that structural changes at the high frequencies and intermediate strains
(b) are unable to reach a steady state within the period, reflecting the
fact that the applied strain is below yielding.

Elliptic Lissajous plots at low strain amplitudes (fig 1) are characteristic
of a linear viscoelastic response. At higher amplitudes the Lissajous curves
deviate from such linear response. There, at low $Pe_{\omega }$ the
structure remains almost isotropic throughout the period (due to Brownian
relaxation) with the Lissajous plot revealing an elastic response around the
strain maximum (stress increases almost linearly with strain) and a
subsequent plastic flow (stress plateau) around zero strain. In contrast at
high $Pe_{\omega }$ the structure is highly distorted and the Lissajous
plots reveal a characteristic stress reduction upon strain reversal as
discussed in the text.

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