  Exclusive $C=+1$ scalar, axial-vector, and tensor quarkonium production in
high-energy electron-proton scattering requires a $C$-odd $t$-channel exchange
of a photon or a three gluon ladder. We derive the expressions for the
corresponding amplitudes. The relative phase of the photon vs. three gluon
exchange amplitudes is determined by the sign of the light-front matrix element
of the eikonal color current operator $d^{abc}J^{+a}J^{+b}J^{+c}$ at moderate
$x$, and is not affected by small-$x$ QCD evolution. Model calculations predict
constructive interference, which is particularly strong for momentum transfer
$|t|\sim 1$~GeV$^2$ where the cross section for $\chi_{cJ}$ production exceeds
that for pure photon exchange by up to a factor of 4. We find that exclusive
$\chi_{cJ}$ electroproduction at the Electron-Ion Collider should occur with
well measurable rates and measurements of these processes should allow to find
an evidence of the perturbative Odderon exchange. We also compute the total
electroproduction cross section as a function of energy and provide first
estimates of the number of $\chi_{cJ}$ events per month at the Electron-Ion
Collider design luminosity.
