Although multidimensional simulations have investigated the processes of double white dwarf (WD) mergers, postmerger evolution only focused on the carbon-oxygen (CO) or helium (He) WD merger remnants. In this work, we investigate for the first time the evolution of the remnants stemming from the merger of oxygen-neon (ONe) WDs with CO WDs. Our simulation results indicate that the merger remnants can evolve to hydrogen- and helium-deficient giants with a maximum radius of about 300 R (circle dot). Our models show evidence that merger remnants more massive than 1.95 M (circle dot) can ignite Ne before significant mass loss ensues, and they thus would become electron-capture supernovae. However, remnants with initial masses less than 1.90 M (circle dot) will experience further core contraction and longer evolutionary time before reaching the conditions for Ne burning. Therefore, their fates are more dependent on mass-loss rates due to stellar winds and thus more uncertain. Relatively high mass-loss rates would cause such remnants to end their lives as ONe WDs. Our evolutionary models can naturally explain the observational properties of the double WD merger remnant IRAS 00500+6713 (J005311). As previously suggested in the literature, we propose and justify that J005311 may be the remnant from the coalescence of an ONe WD and a CO WD. We deduce that the final outcome of J005311 would be a massive ONe WD rather than a supernova explosion. Our investigations may be able to provide possible constraints on the wind mass-loss properties of the giants that have CO-dominant envelopes.