In more than one video, I've heard Jordan Peterson speak of a right/left divide in what strikes me as simplistic terms. For him, the right is characterized as preferring the efficiency of hierarchy, while the left prefers to advocate for those at the bottom. I've also heard him say that no matter what system of individual value you set up (whether based on IQ, on skill, on education, on the "purity" of "political correctness"...) any system you set up will naturally evolve into a hierarchy. We have, of course, seen this in every government system in recorded human history. The end of history described by Marx and Engels was, according to this view (and all human experience) pure fantasy. We can recognize the truth of this, in part, by acknowledging the point that all systems develop a hierarchy.
That said, I think it's incorrect to suggest that "the right" prefers hierarchy because of its efficiency. The most, it seems to me, that we can say is that the various branches of "the right" acknowledge that hierarchy is natural. This is true even of so-called "radical right" socialist systems, such as Nazism or Fascism. It's also true of almost every person on the left (not just the "radical right" leftists!), no matter of what stripe, because to believe that hierarchy is not, in any fundamental sense, natural, is to live in a fantasy world. The best evidence of this is that every socialist system (especially those attempted on a national scale) has had a hierarchy.
Jesus, himself, acknowledged hierarchy, even hierarchy in heaven. He also pointed out, by way of deepening our appreciation of the beauty of sacrificial love, that anyone who wishes to be greatest in the kingdom of heaven must be the servant of all on earth. (Cf. Matthew 20:26-27, Mark 9:35, John 13:1-17.) His focus was on inculcating a spirit of servant leadership. Thus, for Jesus, the Son of God, our position in heaven will be determined by the height of sacrificial love we internalize (in the sense of developed and/or infused virtue) here on earth.
Even saying that what characterizes the "left" is advocacy for those at the bottom is a bit simplistic because so-called "leftists" almost always discount the personhood of the unborn and refuse to acknowledge that they have the right to life, whereas a major component of the so-called right is the social conservatives who do acknowledge that right and who often go to great lengths to defend that right. The same is true for those at the end of life, with the understanding that life, itself, is inherently valuable, even a life of suffering, though nearly everyone will admit that efforts to alleviate suffering are admissible, even, in some sense, laudatory, in most practical cases. (Acknowledging that suffering is generally a sign that something is wrong, and the best way to alleviate that suffering is to right that wrong. Thus, in medicine, suffering is generally understood to be a symptom of an underlying malady, and curing or repairing the malady, if possible, is the best way to alleviate the suffering.) One can even go further and recognize that certain systems, whether well-meaning or not, end up trapping people at the bottom by devaluing human striving and over-regulating the economy to the advantage of larger entities, including unions as well as corporations. Furthermore, any serious student of history will acknowledge that the history of progressivism is intertwined with the history of eugenics, and that so-called "progressive" movements have sponsored mass sterilization, forced (or, at any rate, subsidized) abortions, and other forms of "birth control". One of the most recent examples of this is the mania behind eliminating Down Syndrome from the gene pool (a fool's errand if there ever was one, even under the assumption that the presence of people with this syndrome in the population is somehow "bad" for society), a thoroughly dehumanizing enterprise.
Personally, as I have indicated in other posts on this blog (and elsewhere!), I prefer to decompose political differences in a multi-dimensional analysis. I'm happy to acknowledge that I find this not only to be the most accurate fundamental way to distinguish differences, but that it is a method which appeals to my mathematical background. Peterson's background, by contrast, is in psychology (his Ph.D. work) as well as political science, history and philosophy. (As you may or may not have observed, I've done work in these areas as well.) From his perspective, in order to engage people in a rational discussion, it helps to simplify the characterization of "right" vs. "left" that most people find congenial. I can sympathize with that, but I've found that it's easier to make progress in these kinds of discussions if people are capable of acknowledging and working with the multi-dimensionality of politics.
Furthermore, things like hierarchy and power are inherently multi-dimensional in human societies. Think about it!
That said, I am happy to acknowledge that most people who think in terms of "right" vs. "left" will be open to a dialog predicated on the distinction that Peterson makes, even if it means they abandon, partially, their preferred way of thinking of the differences for the sake of argument. The most rational reason for doing so, it seems to me, is that there is a natural kind of dynamic in human society that involves the interplay between the evolution of hierarchy and the periodic movements that disrupt these hierarchies that are motivated by concern for people at the bottom, or by loathing of the people at the top, or, what is perhaps the most frequent case, both.
Judaism, Christianity and Mahayana Buddhism have fundamental principals and rules or laws that involve a focused care for the dispossessed and downtrodden. (No doubt there are other religions that have comparable concerns and values, but these are the examples I've had occasion to study.) It seems to me (and you are free to dismiss this as biased, if you like) that Jesus' articulation of servant-leadership is the most profound and productive way of integrating this concern into human society, particularly if we think of it not only in terms of individual leadership, but also in terms of the collective leadership of groups. (See, for example, the historically Catholic notion of "preferential option for the poor and vulnerable".) The stability of society is vastly improved when this idea is integrated into our social DNA, along with other natural human tendencies that support cooperation, inventiveness and enterprise.
By contrast, I find the use of terms like "left" vs. "right" (especially "radical left" vs. "radical right") to be inherently distancing and, as a consequence, destabilizing.