With all the crazy talk about George Church and an adventurous young woman conspiring to bring back Neandertals, I do think it is important to keep in mind that
we can bring back an individual with a predominantly Neandertal genome in a very old fashioned manner: controlled breeding.
The most humane and viable manner in which you might do this is simply start a religion in a Bene Gesserit fashion where the prophesied Kwisatz Haderach is a Neandertal. Over the generations by selecting individuals within the population (which could draw in converts) enriched for Neandertal ancestry to mate assortatively one could slowly increase the proportion of that ancestral component. The population would become more and more "Neandertal," probably to the point of being phenotypically distinctive in a dozen generations (even a minority of non-modern human ancestry is probably significant, just as many individuals who are 3/4 European and 1/4 African still exhibit features of their minority heritage). One could apply the same logic to the Denisovans. I think this obviates the bioethical concerns, as a natural group of Neandertals would emerge over time in an organic fashion. At least in a genomic sense (though likely there will always be "gaps" in the reconstructed genome; for example the mtDNA). One might object that one can't bring back Neandertal culture, but it strikes me as patronizing and peculiar to presume that Neandertals (in a genomic sense) should exhibit a biocultural integrity that we don't expect from the descendants of anatomically modern humans. No doubt ever generation young Neo-Neandertals would leave the cultish environs of their natal sect. Perhaps there would be apostate Neo-Neandertals making the hajj. Kissing the ring of the Pope in Rome. Retreating to Fiji to paint the native modern humans. You might think this is a farcical suggestion, but why not? Today we live in the world of individual self-actualization and hedonic utilitarians. In short, we live as individuals to wallow in our own pleasure. Why not look deep into the future, perhaps beyond our own mortality? The resurrection of assimilated hominin lineages would be a noble artistic experiment. Even if the revived lineages are clearly different in fundamental ways, it would add to the diversity of the world. Is in a sin that they would "think different"?