Stable oxygen sites on a PdO film over a Pd(100) thin structure with a surface unit cell are determined using the first-principles electronic structure calculations with the generalized gradient approximation. The adsorbed monatomic oxygen goes to a site bridging two twofold-coordinated Pd atoms or to a site bridging a twofold-coordinated Pd atom and a fourfold-coordinated Pd atom. Estimated reaction energies of CO oxidation by reduction of the oxidized PdO film and N₂O reduction mediated by oxidation of the PdO film are both exothermic. Motion of the adsorbed oxygen atom between the two stable sites is evaluated using the nudged elastic band method, where an energy barrier for a translational motion of the adsorbed oxygen may become ~0.45 eV, which is low enough to allow fluxionality of the surface oxygen at high temperatures. The oxygen fluxionality is allowed by the existence of twofold-coordinated Pd atoms on the PdO film, whose local structure has a similarity to that of Pd catalysts for the Suzuki–Miyaura cross-coupling. Although NO_x (including NO₂ and NO) reduction is not always catalyzed by the PdO film only, we conclude that continual redox reactions may happen mediated by oxygen-adsorbed PdO films over a Pd surface structure, when the influx of NO_x and CO continues, and when the reaction cycle is kept on a well-designed oxygen surface.