  We experimentally investigate the lattice-induced light shift by the
electric-quadrupole ($E2$) and magnetic-dipole ($M1$) polarizabilities and the
hyperpolarizability in Sr optical lattice clocks. Precise control of the axial
as well as the radial motion of atoms in a one-dimensional lattice allows
observing the $E2$-$M1$ polarizability difference. Measured polarizabilities
determine an operational lattice depth to be $72(2) E_R$, where the total light
shift cancels to the $10^{-19}$ level, over a lattice-intensity variation of
about 30%. This operational trap depth and its allowable intensity range
conveniently coincide with experimentally feasible operating conditions for Sr
optical lattice clocks.
