Researching for the other question, I found many interesting sources about the compatibility (or incompatiblity) of transhumanism and conservatism.
First, this very interesting article from Ilia Stambler, a supporter of life extension initiatives, Life extension – a conservative enterprise? Some fin-de-siècle and early twentieth-century precursors of transhumanism:
The beginning of the modern period in the pursuit of radical human enhancement and longevity can be traced to fin-de-siècle/early twentieth-century scientific and technological optimism and therapeutic activism. The works of several authors of the period – Fedorov, Stephens, Bogdanov, Nietzsche and Finot – reveal conflicting ideological and social pathways toward the goals of human enhancement and life extension. Each author represents a particular existing social order, and his vision of human advancement may be seen as a continuation and extension of that order. Therefore, the pursuit of life extension may be considered a fundamentally conservative (or conservationist) enterprise.
(...) in the authors under consideration, the goal of life extension has been associated with a striving for stability and equilibrium, desiring to stabilize and thus perpetuate the current state of the body or personality, and the present social system. In this sense, life-extensionism may be a fundamentally conservative (or conservationist) enterprise. Therefore, the impression that life-extensionism represents a form of utopianism, a fringe or revolutionary movement, or an advocacy of a radical change of the human nature – should be rejected or accepted only with profound reservations. Historically, the proponents of radical life extension may have envisioned no greater change to human nature than the extent to which maintenance of an ancient edifice changes the nature of that edifice. The life-extensionists may indeed have strived for a perfected society, which one might call a “utopia,” but that “utopian” society, they hoped, would uncannily resemble the one they lived in, with all or most of its institutions intact and all the near and dear ones alive and around. The life-extensionist movement may have been profoundly anti-revolutionary, if only for the simple reason that opposing the existing social system would nullify public support of longevity research.
Still, there might be a point of collision between “traditional” life-extensionism and “transhumanism” or “singularitarianism.” The singularitarians (such as Kurzweil) will have us believe that human-level artificial intelligence and man-machine synergy are inevitable, given the current trend of accelerating development of information technology. It is this technology that Kurzweil anticipates will help us find effective life-extending means through data mining, will provide a backup for our personal traits for future “uploading” or “fine-tuning,” and will direct the repair of our body. But at some stage, Kurzweil believes, the machines will succeed us entirely: in the best scenario our memories will be a component of machine intelligence, and in the worst the machines will entirely supersede the human race. Such a level of change may be too great to accept for a “traditional life-extensionist” wanting to be around in the same (or very similar) body and environment (i.e., it could induce an incapacitating “future shock”).
So, for this author, life extension (one of the most important goals of most transhumanist groups) is actually a conservative and not a revolutionary entreprise - because they both have the goal to conserve social order. However, singularitarianism ends to be a too radical change for a conservative
Quoting and discussing the Ilia Stambler article, there is this post, Transhumanism as a Grand Conservatism, where the author, G. Stolyarov II (a self professed libertarian transhumanist), claims that
I am mostly not a conservative in the American or even European political sense, but I am conservative in the sense of seeking to preserve and build upon the achievements of Western civilization – including the development of its logical implications for future decades and centuries. Technological progress and the achievement of indefinite life extension are very much the direct extrapolation of the desire to preserve the historical achievements that enable our unprecedented quality of life today. I have no ambitions to have my mind “uploaded,” to lead a non-biological existence, or “merge” my mind with anyone else’s. If I obtain indefinite life, I will spend it indefinitely looking the way I do (while remedying any flaws) and focusing on the perpetuation of my family, property, esthetic, and activities – all the while learning continuously and becoming a better (and more durable) version of the person I already am.
Cultural and historical preservation is also a major but seldom appreciated implication of transhumanism. By living longer and remaining in a youthful state, specific individuals would be able to create and refine their skills to a much greater extent. Imagine the state of classical music if we could have had hundreds of years for Mozart and Beethoven to compose – or the state of painting if Leonardo, Vermeer, or David had lived for centuries. Every time a creator dies, an irreplaceable vision dies with him. Others might emulate him, but it is not the same – for they do not have his precise mind. They can replicate and absorb into their own esthetic what he already brought into this world, but they cannot foresee the new directions in which he would have taken his work with more time. Each individual is precious and irreplaceable; the loss of each individual is the loss of a whole universe of memories, ideas, and possibilities. Transhumanism is a grand conservatism – an ambition to conserve people – to put an end to all such senseless destruction and to keep around all of the people who build up and beautify our world. The proto-transhumanist Nikolai Fedorov (one of those Christian transhumanists who ought to be much more prevalent among the Christians of today) even took this idea to the point of proposing an ultimate goal to physically resurrect every person who has ever lived.
So, in a sense of preserving a culture and the historic achievements of a civilization, conservatism and the life-extensionist/immortalist strand of transhumanist, again, have a sinergy of goals. However, things like mind uploading or other radical changes in what the individual is are not desired by the conservative-leaning libertarian author.
Another discussion can be found in the blog post I quoted in the question, where the democratic transhumanist author, "citizencyborg", claims not only that there are few transhumanist conservatives but
The most important political challenge for a "conservative transhumanism" is articulating how a transhumanist can support the restrictions on bodily autonomy, cognitive liberty and civil rights that are integral to conservative policies. For me, both a rejection of fetal personhood and a commitment to adult women's right to control their own body, requiring in turn unrestricted access to abortion, is pretty central to transhumanist values. Articulating the conservative transhumanist case for more restrictions on abortion, or conversely why conservatives should defend abortion rights, would be pretty important. Same for gay rights, stem cells, the Drug War, prayer in schools, creationism, and a lot of the other defining issues of modern US conservatism.
We could imagine the consensual use of neurotechnologies by individuals in traditional communities to suppress desire for premarital sex, homosexuality and adultery, to accentuate religious belief (especially if there is God Gene) and other "virtues" such as charity, patriotism, honesty and so on. I actually make that argument, on behalf of empathy and morality, at the end of Citizen Cyborg.
So, for the author, a transhumanist conservative needs to not be a conservative (or, better saying, a religious conservative) in almost all moral and religious hot-topic political issue, since he thinks that a transhumanist needs necessarily to be pro-choice and pro-LGBT rights, among other progressist (or libertarian) social positions. However, he still can support conservative values through consensual use of technologies that can help a person to live according to what a (religious) conservative believe is a moral and correct life.
And, in the comments, a supposedly self-professed conservative transhumanist, Rev. Thomas Scott Painter, have a different opinion about life extension than Ilia Stambler and G. Stolyarov II:
Insure modest gains for the next generation, having great faith in your eventual resurrection rather than selfishly gambling everything on yourself and risking the survival of the human race which is supposed to play a role in your eventual resurrection. We don't want to die, but most of us don't see much hope of extending our maximum lifespans during our lifetimes, so we are highly involved in propagating our memes, ensuring the remaining remnant of humanity will take part in the resurrection which prepares us for the day in which we are judged by the superintelligence which made us human because he wanted human things to be done. Our main purposes in life is to please the superintelligence by giving him all glory and honor and power for no other reason than he is who he is. I like to call him God but all names are his because he's responsible for everything. :-)
We tend to focus on more immediate threats to the survival of humanity, but transhumanism simmers¹ in the background if you look for it.
¹(I swear I read "sinners" the first time I read this phrase!)
Basically, he is skeptical about life extension, but wants to "propague our memes" (I suppose this means propague Christian religion) to help the remnant (non-Christian?) of mankind to ensure salvation in the judgment day. This is an interpretation of transhumanism suited to work with the Religious Right, a group that intuitively seems to be more aversive to the transhumanist ideology.
This blogger, Pop-Bioethics, add similar arguments to "citizencyborg" post:
The problem of politics in transhumanism is that a movement that aims to expand and grow the human race, in totem, cannot be conservative in any sense. It must, by definition, be progressive and liberal in the apolitical meaning of the words. Concepts that ‘conservatives,’ in particular social conservatives, hold dear, be it strong family and religion or restricted sex and drugs, stand in direct opposition to transhumanism. One cannot be conservative and transcendant at the same time. It is a paradox.
The goal is to make it possible to be conservative within transhumanism, that is: one could be a Catholic, member of a nuclear family, drug-free cyborg. Or not. The option to be either, the choice and ability to do both, means that conservative transhumanism must articulate a postition that supports conservative behavior and values (monogamy, religiousness) but allows very extreme transgressions of those behaviors and values (polyamory, atheism). Thus, even conservative transhumanists, in order to prevent hypocrisy, must have such a high level of tolerance that, to bioconservatives and traditional conservatives alike, they will appear liberal and left-of-center.
As "citizencyborg", this author claims that a transhumanist conservative can't hold conservative positions in most moral issues; this can be understood as that a transhumanist needs to support what could be stated as the freedom to control and alter his own body, or Morphological Freedom. However, it's very questionable (to say the least) that to accept that others can be atheist is a "very extreme transgressions of those [conservative] behaviors", since except for the more extremist most conservative are not for forbidding atheism.
James Hughes, a leading member of transhumanist community and executive director of the Institute for Ethics and Emerging Technologies (and a democratic transhumanist too), states that
Strictly, I think you could believe in biblical inerrancy, be a staunch
nationalist, and be opposed to homosexuality and abortion, and
nonetheless be an enthusiastic supporter of our rights to use human
enhancement technologies. Any such Christian fundamentalist and
nationalist transhumanist is welcome in the WTA [World Transhumanist Association]. But I understand why not many find this to be their home.
As to being opposed to taxation and being opposed to regulation, we have
little problem. We are generally perceived as libertarians, and in fact
1/3 of our membership are one kind of libertarian or another.
As to foreign policy, I don't think there is a strong correlation
between, for instance, opposition to the United Nations, or support for
Pax Americana, and transhumanism. So I make no prediction on that score.
(...) I don't think either of those [being for a strong US military and being opposed to the social welfare state] views are incompatible with
This prominent democratic transhumanist disagree with his aforementioned collegues, and don't see opposition between being pro-life and "an enthusiastic supporter of our rights to use human enhancement technologies".
So, TLDR conclusion: for some authors, life extensionism is a conservative entreprise because it wants to conserve the existence of individuals, while helping to preserve culture and historical achievements of a civilization and strengthen social stability, while more radical strands of transhumanism are less compatible with conservative goals.
However, for some authors more aligned with democratic transhumanist views, a transhumanist needs to uphold the freedom to use and modify your bodily free (including abortion), and (for these same authors), conservatives are by default against this freedom. A conservative transhumanist is then required to, while personally not using these rights, allow others to enjoy them. But this position is not consensual, and a transhumanist can at least be anti-abortion and opposed to LGBT movement's agenda.