I think that the left-right dichotomy is inherently flawed. A lot of what I believe might be considered “right-leaning” or “left-leaning,” but I cannot say that I prescribe to either sort of ideology fully or with any fidelity.
I will always be opposed to any view with a pervasive “moral” authority, and both the so-called left and right are obsessed with their own versions of this. The problem we run into is the false supposition that beliefs can be categorized on a spectrum spanning right to left (or, even more liberally, a spectrum spread across two dimensions). It has been a ridiculous notion from its inception, whenever that might have been.
Building one’s identity (another silly notion, in general—identity itself being a frivolous construct that functions only as a fulcrum for the extortion of social power) upon a supposed spectrum is likewise ridiculous. You can be conservative or liberal, or anything, really. But those beliefs do not exist in a linear or planar dimension. They are so far removed from each other that one cannot fathom sliding incrementally from one to the next.
And to each respective party, “left” and “right,” the other can be demonized as evil, even without full comprehension of the other. It’s all just so damned tribalistic and silly.
I will tell you why atheism, regardless of the horse on which it rides, is a step toward progress: it is because belief in fantasy is an opiate that numbs us and stymies progress toward improving the welfare of all.