We investigate the nature of valency and bonding in the highly unusual O2PR3 "phosphadioxirane" species recently isolated as an intermediate in the reaction of O-1(2) with organic phosphines PR3. Commonly, this species is depicted as a Lewis structure with five bonds at the phosphorus center, suggesting hypervalent involvement of extra-valence d-orbitals in the hybridization. However, nonhypervalent bonding patterns, such as open zwitterionic peroxides or R2PO2+R- ion pairs, could also achieve the observed hypercoordination. In the present work, we employ ab initio and hybrid density functional calculations with theoretical analysis by means of Natural Resonance Theory (NRT) and Natural Bond Orbitals (NBOs) to investigate the role of valence shell expansion versus nonhypervalent ionic resonance in phosphadioxiranes. We find that true hypervalency is relatively negligible in phosphadioxiranes, and hypercoordination is instead achieved through both conventional linear Pimentel-Rundle three-center, four-electron (3c/4e) hyperbonding as well as an unprecedented cyclic form of 3c/4e hyperbonding. We examine ramifications and limits of the cyclic hyperbonding phenomenon in analogous carbon compounds and discuss some broader implications of its structural representation and nomenclature.