  Determining the $CP$ property of the Higgs boson is important for a precision
test of the Standard Model as well as for the search for new physics. We
propose a novel jet substructure observable based on the azimuthal anisotropy
in a linearly polarized gluon jet that is produced in association with a Higgs
boson at hadron colliders, and demonstrate that it provides a new $CP$-odd
observable for determining the $CP$ property of the Higgs-top interaction. We
introduce a factorization formalism to define a polarized gluon jet function
with the insertion of an infrared-safe azimuthal observable to capture the
linear polarization.
