  We investigate the azimuthal asymmetries such as $\cos2{\phi_T}$ and Sivers
symmetry for $J/\psi$ and $\pi^\pm$ production in electron-proton scattering,
focusing on scenarios where the $J/\psi$ and the pion are produced in an almost
back-to-back configuration. The electron is unpolarized, while the proton can
be unpolarized or transversely polarized. For the $J/\psi$ formation, we use
non-relativistic QCD (NRQCD), while $\pi^\pm$ is formed due to parton
fragmentation. In this kinematics, we utilize the transverse momentum-dependent
factorization framework to calculate the cross sections and asymmetries. We
consider both quark and gluon-initiated processes and show that the gluon
contribution dominates. We provide numerical estimates of the upper bounds on
the azimuthal asymmetries, as well as employ a Gaussian parametrization for the
gluon transverse momentum distributions (TMDs), within the kinematical region
accessible by the upcoming Electron-Ion Collider (EIC).
