In this paper we perform an amplitude analysis of essentially all published pion and kaon pair production data from two photon collisions below 1.5 GeV. This includes all the high statistics results from Belle, as well as older data from Mark II at SLAC, CELLO at DESY, Crystal Ball at SLAC. The purpose of this analysis is to provide as close to a model-independent determination of the \(\gamma\gamma\) to meson pair amplitudes as possible. Having data with limited angular coverage, typically \(|\cos \theta| < 0.6-0.8\), and no polarization information for reactions in which spin is an essential complication, the determination of the underlying amplitudes might appear an intractable problem. However, imposing the basic constraints required by analyticity, unitarity, and crossing-symmetry makes up for the experimentally missing information. Final state interactions among the meson pairs are critical to this analysis. To fix these, we include the latest \(\pi\pi\to\pi\pi\), \({\overline K}K\) scattering amplitudes given by dispersive analyses, supplemented in the \({\overline K}K\) threshold region by the recent precision Dalitz plot analysis from BaBar. With these hadronic amplitudes built into unitarity, we can constrain the overall description of \(\gamma\gamma\to\pi\pi\) and \(\overline{K}K\) datasets, both integrated and differential cross-sections, including the high statistics charged and neutral pion, as well as \(K_sK_s\) data from Belle. Since this analysis invokes coupled hadronic channels, having data on both \(\gamma\gamma\to\pi\pi\) and \(\overline{K}K\) reduces the solution space to essentially a single form. We present the partial wave amplitudes, show how well they fit all the available data, and give the two photon couplings of scalar and tensor resonances that appear. These partial waves are important inputs into forthcoming dispersive calculations of hadronic light-by-light scattering.