We extend previous work on sum rules for the invariant amplitudes of pion Compton scattering by deriving a complete lowest-order perturbative spectral function - and its leading non-perturbative power corrections - for a specific combination of the two helicities (H-1 + H-2) of this process. Using some properties of a modified version of the Borel transform, we develop a method of calculation of the gluonic corrections which can be easily extended to other similar reactions, such as proton Compton scattering. A preliminary comparison of the new sum rule with the pion form factor sum rule is made.