We study the process of associated photon and jet production, pp̅ →γ+jet+X, using 8.7  fb-1 of integrated luminosity collected by the D0 detector at the Fermilab Tevatron Collider at a center-of-mass energy √s=1.96  TeV. Photons are reconstructed with rapidity |yγ|<1.0 or 1.5<|yγ|<2.5 and transverse momentum pTγ>20  GeV. The highest-pT jet is required to be in one of four rapidity regions up to |yjet|≤3.2. For each rapidity configuration we measure the differential cross sections in pTγ separately for events with the same sign (yγyjet>0) and opposite signs (yγyjet≤0) of photon and jet rapidities. We compare the measured triple-differential cross sections, d3σ/dpTγdyγdyjet, to next-to-leading order (NLO) perturbative QCD calculations using different sets of parton distribution functions, and to predictions from the sherpa and pythia Monte Carlo event generators. The NLO calculations are found to be in general agreement with the data but do not describe all kinematic regions.