Polarized jet anisotropy at the Electron-Ion Collider
Jets provide a powerful probe of the three-dimensional spin structure of the nucleon, a central goal of the Electron-Ion Collider. Yet the observed jet defines an axis that breaks the azimuthal isotropy of soft-gluon radiation, thereby reshaping the very asymmetries used to extract that structure. Using transverse-momentum-dependent (TMD) QCD factorization, we show for the first time that this jet-induced anisotropy imposes a parity selection rule on polarized asymmetries. Expanding the transversely polarized structure functions in harmonics $\cos(n\phi_{qJ})$, where $\phi_{qJ}$ is the angle between the jet and the lepton-jet momentum imbalance, makes this rule explicit: the symmetry of each harmonic is fixed by the parity of $n$, independently of the magnitudes of the harmonic coefficients. For the Sivers function, the canonical left-right asymmetry about the proton spin survives for even $n$ but is replaced by a counterintuitive left-right symmetry for odd $n$. The worm-gear function retains its up-down asymmetry at every harmonic while being left-right symmetric for even $n$ and asymmetric for odd $n$. At EIC kinematics, the higher harmonics studied here are predicted to be individually measurable, providing new observables for the azimuthal dynamics of soft radiation and an essential ingredient in precision extractions of nucleon structure.