  We reconsider Muskhelishvili-Omn\`es (MO) dispersive representations of
photon-photon scattering to two pions, motivated by the very high statistics
results recently released by the Belle collaboration for charged as well as
neutral pion pairs and also by recent progress in the determination of the
low-energy $\pi\pi$ scattering amplitude. Applicability of this formalism is
extended beyond 1 GeV by taking into account inelasticity due to $K\bar{K}$ . A
modified MO representation is derived which has the advantage that all
polynomial ambiguities are collected into the subtraction constants and have
simple relations to pion polarizabilities. It is obtained by treating
differently the exactly known QED Born term and the other components of the
left-hand cut. These components are approximated by a sum over resonances. All
resonances up to spin two and masses up to $\simeq1.3$ GeV are included. The
tensor contributions to the left-hand cut are found to be numerically
important. We perform fits to the data imposing chiral constraints, in
particular, using a model independent sum rule result on the $p^6$ chiral
coupling $c_{34}$. Such theoretical constraints are necessary because the
experimental errors are dominantly systematic. Results on further $p^6$
couplings and pion dipole and quadrupole polarizabilities are then derived from
the fit. The relevance of the new data for distinguishing between two possible
scenarios of isospin breaking in the $f_0(980)$ region is discussed.
