  In our previous work [Phys. Rev. Lett. 128, 182002 (2022)], we introduced a
pioneering observable aimed at experimentally detecting the orbital angular
momentum (OAM) of gluons. Our focus was on the longitudinal double spin
asymmetry observed in exclusive dijet production during electron-proton
scattering. We demonstrated the sensitivity of the $\cos \phi$ angular
correlation between the scattered electron and proton as a probe for gluon OAM
at small-$x$ and its intricate interplay with gluon helicity. This current work
provides a comprehensive exposition, diving further into the aforementioned
calculation with added elaboration and in-depth analysis. We reveal that, in
addition to the gluon OAM, one also gains access to the spin-orbit correlation
of gluons. We supplement our work with a detailed numerical analysis of our
observables for the kinematics of the Electron-Ion Collider. In addition to
dijet production, we also consider the recently proposed semi-inclusive
diffractive deep inelastic scattering process which potentially offers
experimental advantages over dijet measurements. Finally, we investigate
quark-channel contributions to these processes and find an unexpected breakdown
of collinear factorization.
