  We apply three forward light-by-light scattering sum rules to charmonium
states. We show that these sum rules imply a cancellation between charmonium
bound state contributions, which are mostly known from the $\gamma \gamma$
decay widths of these states, and continuum contributions above $D \bar D$
threshold, for which we provide a duality estimate. We also show that two of
these sum rules allow to predict the yet unmeasured $\gamma^\ast \gamma$
coupling of the $\chi_{c1}(1P)$ state, which can be tested at present
high-luminosity $e^+ e^-$ colliders.
