Coherent photoproduction of the neutral pion and eta meson on the deuteron, γd→π 0 ηd, has been experimentally studied at incident photon energies ranging from the reaction threshold to 1.15 GeV. The total cross section demonstrates a rapid rise below 1 GeV. The data are underestimated by the existing theoretical calculations based on quasi-free π 0 η photoproduction on the nucleon followed by deuteron coalescence. At the same time, the data are rather well reproduced by the calculations taking into account the final-state interaction. We have also measured for the first time the differential cross sections: the π 0 η invariant-mass distribution dσ/dMπη, the π 0 d invariant-mass distribution dσ/dM πd , the ηd invariant-mass distribution dσ/dM ηd , and the distribution over the deuteron emission angle dσ/dΩ d in the overall center-of-mass frame. The measured cross section dσ/dΩ d does not exhibit strongly backward-peaking behavior predicted by the calculations. At all incident energies, an increase in dσ/dM ηd near the ηd threshold is observed, which indicates a bound or virtual ηd state resulting from a strong attraction between η and a deuteron. The possibilities of using coherent π 0 η photoproduction on a nucleus to study the η-nuclear interaction are also discussed.