  The high-statistics Belle data on the \gamma\gamma\to\pi^+\pi^- and
\gamma\gamma\to\pi^0\pi^0 reactions have been jointly analyzed. The main
dynamical mechanisms of these reactions for energies below 1.5 GeV have been
revealed. It has been shown that the direct coupling constants of the
\sigma(600) and f_0(980) resonances with a \gamma\gamma pair are small and that
the \sigma(600)\to\gamma\gamma and f_0(980)\to\gamma\gamma decays are
four-quark transitions due primarily to \pi^+\pi^- and K^+K^- loop mechanisms,
respectively. The role of the chiral shielding of the \sigma(600) resonance is
emphasized. The widths of the f_0(980)\to\gamma\gamma and
\sigma(600)\to\gamma\gamma decays averaged over the resonance mass
distributions, as well as the width of the f_2(1270)\to\gamma\gamma decay, are
estimated as (\Gamma_{f_0\to\gamma\gamma})_{\pi\pi}\approx 0.19 keV,
(\Gamma_{\sigma \to\gamma\gamma})_{\pi\pi}\approx 0.45 keV, and
\Gamma_{f_2\to\gamma\gamma}(m^2_{f_2})\approx 3.8$ keV.
