After briefly introducing the Odderon exchange and recalling some previous searches at HERA, I will move on to discussing exclusive chi_c quarkonia productions and argue this could be a discovery channel for the Odderon at the EIC. The Odderon interferes (constructively) with the photon (Primakoff) exchange at the amplitude level, and we find the Odderon dominates after |t| ~ 1-1.5 GeV^2 at the EIC kinematics. The high luminosity of the EIC is enough to produce about dozen chi_c's per month through their radiative J/psi + gamma decay. In the near forward limit I explain how the (spin dependent) Odderon) could be probed through the measurement of the polarization of the axial chi_c1.