  We present results for the charged and neutral pion polarizabilities,
obtained through a dispersive analysis of photon-fusion reactions with two
pions in the final state. This analysis is motivated by current and future
measurements at COMPASS, JLab (Hall D), and the BESIII measurement of
single-virtual photon-fusion reactions in the space-like region, offering
insight into generalized pion polarizabilities. To improve predictions within
an unsubtracted dispersion formalism with only the pion left-hand cut, one
needs to consider the influence of heavier left-hand cuts. We parametrize the
latter using an expansion in a suitably constructed conformal variable,
capturing the analytical structure of the left-hand cuts. The coefficients in
this expansion are determined from the matching with the Lagrangian-based
approach, the Adler zero constraint for $\gamma\gamma \to \pi^0\pi^0$, and a
fit to the cross-section data. The extension to the single-virtual case is also
discussed.
