Looks at history textbooks.
Walks away against the setting sun, smoking a limp cigarette.
A People's History of the United States
The first chapter is about Columbus and includes excerpts from his own journals. Highly recommended.
I think you bring up a super good point, which is phrased in a way to call into question my claim as to why the "cooperative poor don't work together" conclusion. (And by "poor," here I am including all underprivileged economic classes, which, though distinct for a myriad of reasons, have at least the one unifying aspect of economic poverty.)
But the claim is totally supported by evidence and the reason I didn't mention it is that... like... entire books are written on this subject. Seriously, you basically asked me to provide all the evidence for all of human class struggles throughout all civilizations throughout all of history. That's uh... that's not going to happen in a comment on Reddit.
Here's a great sample of some of the things I've read that touch on this topic. I'm including several about the US here since that's the example I used above, but I'm also including a couple that are very much about this topic but aren't US specific, and one that's a good counter example that still proves the point. And for the sake of irony and ease-of-access, I'll provide Amazon links, but you can Google any of these. Literally any of these would serve as a good start on this topic, and I'll contain myself to 10:
I'll add that the Communist Manifesto or anything about Russia's Bolshevik Revolution or the Chinese Communist Revolutions are similar to Citizens or pretty much anything else that delves into class struggles for the French Revolution. They're all examples of when the poor can coordinate, and they're all the same; i.e. "the poverty unifier" is drastically more significant than the sum total of the other "great dividers." (Disclaimer: Yes, I'm fully aware that Russia, France, and China are not ethnically monolithic, but like... read the history. In the places where revolutions sparked, they sparked among large groups of poor that were far less divided among other lines than they were unified by their poverty.)
You're also correct that I didn't really dive into the evidence for why the claim is accurate (I didn't discover this or really even make this claim. I'm just echoing the work of dozens of other historians, philosophers, and economists), so I'm going to rephrase what you're asking as, "So WHY is it so uncommon for the poor to collaborate together effectively?"
In a nutshell, it's a combination three factors:
As to your point of my "poor-rich dichotomy" versus other dichotomies; first, I'd like to point out that no one "aligns themselves." We are aligned by the forces that create us, and attributing any social identity to "personal choice" is both factually incorrect and, more on topic, (not to sound overly dramatic here) an act of violence to insinuate otherwise.
To answer specifically, racism, sexism, homophobia, etc., though not born from, are all collateral damage of, and exacerbated by, classism. Equitable partitioning of resources (including real estate) and equitable participation in education, politics, and other power structures absolutely diminishes the foundations where any of that other crap can germinate. Hell, you can literally look at the effect that just better education and urban diversity have on the ability not only to collaborate, but for humans to literally "miscegenate" (to borrow a super fucking racist word for, hopefully, a better purpose).
TL;DR: Classism is really bad, and being poor is so devastating that it's really hard to accomplish anything when you're poor, especially if you've been denied access to education.