  The intermediate quarkonium state $\chi_2(^3P_2)$ in electron-positron
annihilation to proton and antiproton as well as in antiproton-proton
annihilation to electron and positron can produce backward-forward asymmetry,
when populated through two photon exchange. We use the dispersion relation
method, which permits to express the asymmetry in terms of partial widths of
quarkonium decay. The asymmetry dependence on the center of mass energy in the
range near the resonance is presented. The comparison with a similar effect in
these reactions with the neutral $Z$-boson in the intermediate state is given.
We show that these effects are $\le 10^{-3}$. The main source of asymmetry is
of pure QED origin ($\sim 10^{-2}$) which arises from the interference between
initial and final state real photon emission.
