  We explore connections between high energy QCD spin physics and $CP$-odd
scalar gluonic operators $\tilde{F}^{\mu\nu}F_{\mu\nu}$ and
$\tilde{F}_{\mu\nu}F^{\mu\alpha}F^{\nu}_{\alpha}$, the latter being called the
Weinberg operator in the context of the nucleons' electric dipole moment. We
first introduce the twist-four generalized parton distribution (GPD) associated
with the topological operator $F_{\mu\nu}\tilde{F}^{\mu\nu}$. This has
interesting applications in spin physics which go beyond the standard framework
in terms of twist-two and twist-three distributions. In the second part, we
show that the off-forward matrix element of the Weinberg operator is
proportional to a certain twist-four correction to the $g_1$ structure function
in polarized deep inelastic scattering.
