  Starting from hyperbolic dispersion relations, we derive a system of
Roy--Steiner equations for pion Compton scattering that respects analyticity,
unitarity, gauge invariance, and crossing symmetry. It thus maintains all
symmetries of the underlying quantum field theory. To suppress the dependence
of observables on high-energy input, we also consider once- and
twice-subtracted versions of the equations, and identify the subtraction
constants with dipole and quadrupole pion polarizabilities. Based on the
assumption of Mandelstam analyticity, we determine the kinematic range in which
the equations are valid. As an application, we consider the resolution of the
$\gamma\gamma\to\pi\pi$ partial waves by a Muskhelishvili--Omn\`es
representation with finite matching point. We find a sum rule for the
isospin-two $S$-wave, which, together with chiral constraints, produces an
improved prediction for the charged-pion quadrupole polarizability
$(\alpha_2-\beta_2)^{\pi^\pm}=(15.3\pm 3.7)\cdot 10^{-4} {\rm fm}^5$. We
investigate the prediction of our dispersion relations for the two-photon
coupling of the $\sigma$-resonance $\Gamma_{\sigma\gamma\gamma}$. The
twice-subtracted version predicts a correlation between this width and the
isospin-zero pion polarizabilities, which is largely independent of the
high-energy input used in the equations. Using this correlation, the chiral
perturbation theory results for pion polarizabilities, and our new sum rule, we
find $\Gamma_{\sigma\gamma\gamma}=(1.7\pm 0.4)\,{\rm keV}$.
