New high-statistics Belle data on the reaction \gamma\gamma\to\pi^0\eta are analyzed to clarify the two-photon production mechanisms and the nature of the a_0(980) meson. The obtained solution for the amplitude \gamma\gamma\to\pi^0\eta is consistent with the chiral theory expectation for the \pi\eta scattering length, with the strong coupling of the a_0(980) to the \pi\eta, K\bar K, and \pi\eta' channels, and with a key role of the rescattering mechanisms a_0(980)\to(K\bar K+ \pi^0\eta+\pi^0\eta')\to\gamma\gamma in the a_0(980)\to\gamma\gamma decay. Such a picture argues in favor of the q^2\bar q^2 nature of the a_0(980) meson and is in agreement with the properties of its partners, the \sigma_0(600) and f_0(980) mesons, in particular, with those that manifest themselves in \gamma\gamma\to\pi\pi. An important role of the vector meson exchanges in the formation of the nonresonant background in \gamma\gamma\to\pi^0\eta is also revealed. The preliminary information on the reaction \pi^0\eta\to\pi^0\eta is obtained.