  In twisted bilayer graphene (TBLG), extremely small deviations from the magic
twist angle $\theta_m{\approx}1.08^\circ$ change its electronic structure near
the Fermi level drastically, causing a meV-wide flat band to appear or
disappear. In view of such sensitivity to minute structural deformations, we
investigate the combined effect of shear and atomic relaxation on the
electronic structure. Using precise experimental data for monolayer and bilayer
graphene as input in a simplified formalism for the electronic structure and
elastic energy, we find TBLG near $\theta_m$ to be unstable with respect to
global shear by the angle $\alpha{\approx}0.08^\circ$. In TBLG, the effect of
shear on the electronic structure is as important as that of atomic relaxation.
Under optimum global shear, calculated $\theta_m$ is reduced by $0.04^\circ$
and agrees with the observed value.
