  We derive new Lorentz Invariance and Equation of Motion Relations between
twist-three Generalized Parton Distributions (GPDs) and moments in the parton
transverse momentum, $k_T$, of twist-two Generalized Transverse
Momentum-Dependent Distributions (GTMDs), as a function of the parton
longitudinal momentum fraction $x$. Although GTMDs in principle define the
observables for partonic orbital motion, experiments that can unambiguously
detect them appear remote at present. The relations presented here provide a
solution to this impasse in that, e.g., the orbital angular momentum density is
connected to directly measurable twist-three GPDs. Out of 16 possible Equation
of Motion relations that can be written in the T-even sector, we focus on three
helicity configurations that can be detected analyzing specific spin
asymmetries: two correspond to longitudinal proton polarization and are
associated with quark orbital angular momentum and spin-orbit correlations; the
third, obtained for transverse proton polarization, is a generalization of the
relation obeyed by the $g_2$ structure function. We also exhibit an additional
relation connecting the off-forward extension of the Sivers function to an
off-forward Qiu-Sterman term.
