Although multidimensional simulations have investigated the processes of double WD mergers, post-merger evolution only focused on the carbon-oxygen (CO) WD or helium (He) WD merger remnants. In this work, we investigate for the first time the evolution of the remnants stemmed 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 maximum radius of about 300Rsun. Our models show evidence that merger remnants more massive than 1.95Msun can ignite Ne before significant mass-loss ensues, and they thus would become electron-capture supernovae (ECSNe). However, remnants with initial masses less than 1.90Msun will experience further core contraction and longer evolutionary time before reaching at 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 an 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 which have CO-dominant envelopes.

22 pages, 10 figures, 8 tables, accepted for publication in ApJL