Advances in nuclear power reactors include the use of mixed oxide fuel, containing uranium and plutonium oxides. The high-temperature behaviour and structure of PuO_2-x above 1,800 K remain largely unexplored, and these conditions must be considered for reactor design and planning for the mitigation of severe accidents. Here, we measure the atomic structure of PuO_2-x through the melting transition up to 3,000 ± 50 K using X-ray scattering of aerodynamically levitated and laser-beam-heated samples, with O/Pu ranging from 1.57 to 1.76. Liquid structural models consistent with the X-ray data are developed using machine-learned interatomic potentials and density functional theory. Molten PuO_1.76 contains some degree of covalent Pu-O bonding, signalled by the degeneracy of Pu 5f and O 2p orbitals. The liquid is isomorphous with molten CeO_1.75, demonstrating the latter as a non-radioactive, non-toxic, structural surrogate when differences in the oxidation potentials of Pu and Ce are accounted for. These characterizations provide essential constraints for modelling pertinent to reactor safety design.