We present a detailed description of the calculation of radiative-recoil corrections to the ground-state hyperfine splitting in muonium and positronium to order α2EF, the results of which were previously reported. All these corrections involve two-photon exchanges with one-loop radiative corrections to either a photon or a lepton. The QED vacuum polarization corrections are evaluated completely analytically to the order of interest. The hadronic contributions are estimated and found to be very small. As a preliminary to the lepton line calculation, a compact expression is derived for the radiative correction to such lines. This factor is then applied in a number of different contexts: the recalculation of the old nonrecoil result, which is known analytically; the analytic evaluation of terms of order α2(memμ)ln(mμme)EF, which arise from the electron leg; and the nonlogarithmic terms from both lines, which require numerical calculations. The muonium results are νμ+e−=(απ)2(memμ)[−2ln2(mμme)+1312ln(mμme)+18.18±0.58]EF and those for positronium are νe+e−=α2(−1.788±0.004)EF, where EF for positronium does not include the annihilation contribution.