Cutting against the grain, I would classify atheism with the left-wing in that atheism as it exists as a mainstream phenomenon fights for overturning traditional ways of thinking about the relationship between man and society. It's true that atheism is not directly political and it is very possible in principle to be a right-wing atheist, but I cannot think of an example of atheism intersecting with politics where the result hasn't been an active effort to push the Overton Window to the left.
Not a bad example, though I will note in my first sentence I did specify "mainstream". At the risk of being pedantic, Rand has largely gotten serious backing (outside of crank bookworm circles) only from conservative leaders who considered themselves to be Christian. This phenomenon is quite different from something like the October Revolution or New Atheism where the intellectual leaders and footsoldiers of the movement are all atheists.
You could certainly argue many of the Nazis fit your view. While their views are considered ridiculous, many saw Christianity as a "Jewish religion" that invaded Europe and dampened a previously strong and warrior-like "Aryan" (i.e., Indo-European) spirit. Many of the Nazis like Göring saw religion as useless and yet saw what they were advocating for as a return to some type of cultural paradigm that even predated Christianity ... arguably, that is extremely culturally reactionary.
P.S. Before someone comes at me with the tired line that the Nazi movement in any way "supported Christianity" because (A) most of its leading figures had at least some vaguely Christian upbringing in an age where nearly 90% of Europe was Christian or (B) that they attempted to cooperate or at least not outright attack the major German churches that a vast majority of the people belonged to ... that's a terrible argument. I highly recommend the book The Nuremberg Interviews by Leon Goldensohn and Robert Gellately, which is available on Audible. It contains hours and hours of actual interviews with all of the Nuremberg Trials defendants while they were imprisoned, as well as 14 prominent Nazis who were incarcerated as witnesses to the main war crimes trials and would be on trial themselves later. In their OWN words, they make it quite clear their regime was tacitly hostile to religion with a plan to replace it as soon as they could at best and outright combative toward it at worst. Additionally, while some of the men refound faith before their death sentences (they were each allowed time with either a Lutheran pastor or Catholic priest), the vast majority claimed to only have any sort of religious affiliation again once on death's door.
You definitely had Nazis who were basically atheists, but they ran a wide gamut and their religious bearings in most cases I am aware of had less to do with rejecting religion in the manner of a Nietzsche or a Dawkins and more to do with rejecting Christianity per se. A particularly interesting (and influential) case is Rosenberg, who was anti-Christian but wanted to replace it with a new religion that would be founded in a theory of German-Nordic supremacy. Perhaps he was dishonestly leveraging the concept of religion and did not believe in it one whit, but if that is the case it seems if anything to bolster the identification of atheism with the left. If we suppose he was a closeted atheist then his development of a racial mythology to replace traditional Christianity seems to be a statement that he did not believe atheism was in fact compatible with the ideological aims of the Nazi regime and that a far-right movement needed a different theological basis.
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