The gas-phase Fe+-mediated oxidation of acetylene by N2O on both sextet and quartet potential energy surfaces (PESs) is theoretically investigated using density functional theory. Geometries and energies of all the stationary points involved in the catalytic reaction are located. For the catalytic cycles, the crucial step is the initial N2O reduction by Fe+ to form FeO+, in which a direct O-abstraction mechanism is located on the sextet PES, whereas the quartet pathway favors a N-O insertion mechanism. Spin inversion moves the energy barrier for this process downward to a position below the ground-state entrance channel. The second step of the catalytic cycles involves two mechanisms corresponding to direct hydrogen abstraction and cyclization. The former mechanism accounts for the ethynol formation with the upmost activation barrier below the entrance channel by about 5 kcal/mol. The other mechanism involves a "metallaoxacyclobutene" structure, followed by four possible pathways, i.e., direct dissociation, C-C insertion, C-to-O hydrogen shift, and/or C-to-C hydrogen shift. Among these pathways, strong exothermicities as well as energetically low location of the intermediates suggest oxidation to ketene and carbon monoxide along the C-to-C hydrogen shift pathway is the most favorable. Reduction of the CO loss partner FeCH2+ by another N2O molecule constitutes the third step of the catalytic cycles, which contains direct abstraction of O from N2O giving OFeCH2+, intramolecular rearrangement to form Fe+-OCH2, and nonreactive dissociation. This reaction is also energetically favored considering the energy acquired from the initial reactants.