  Symmetries essentially provide conservation rules in nonlinear light-matter
interactions, that facilitate control and understanding of photon conversion
processes or electron dynamics. Since anisotropic solids have rich symmetries,
they are strong candidate to control both optical micro- and macroscale
structures, namely spin (circular polarization) and orbital angular momentum
(spiral wavefront), respectively. Here, we show structured high harmonic
generation linked to the anisotropic symmetry of a solid. By strategically
preserving a dynamical symmetry arising from the spin-orbit interaction of
light, we generate multiple orbital angular momentum states in high-order
harmonics. The experimental results exhibit the total angular momentum
conservation rule of light even in the extreme nonlinear region, which is
evidence that the mechanism originates from a dynamical symmetry. Our study
provides a deeper understanding of multiscale nonlinear optical phenomena and a
general guideline for using electronic structure to control structured light,
such as through Floquet engineering.
