  High-harmonic generation (HHG) offers an all-optical approach to discern
structural symmetries through its selection rules and probe topological phases
with its spectral signatures. Here we develop a universal theoretical framework
-- the Jones matrix formalism -- establishing the fundamental relationship
between pulse-crystal shared symmetries and HHG selection rules. Applying this
to the Weyl semimetal (WSM) material TaAs, shows that the anomalous harmonics
excited by linearly and circularly polarized pulses are governed respectively
by the shared twofold and fourfold rotational symmetries of laser pulses and
lattice, rather than the topology of Weyl nodes. The common observables of HHG,
including intensity, circular dichroism, ellipticity dependence, and
carrier-envelope phase dependence, are not found to carry a signature of the
Weyl cones. This insight into TaAs can be extended to other WSMs, indicating
that HHG is not a particularly effective tool for investigating the topological
features of WSMs. However, the Jones matrix formalism lays the groundwork for
both HHG probing crystal symmetry and controlling harmonic polarization states.
