This is one of the stranger books I've read in my time. Swastika Night and The End of This Day's Business were deeply philosophical, but they were airport novels[1] compared to Proud Man which is...
I have no idea what to even say about this book.
The narrator, a sexless human from, presumably, the far future, has no name that they are willing to give to subhumans (read: us), but goes by Verona (clever, Burdekin). They arrive in our time, and live both as a man and as a woman, exploring and commenting on gender roles and subhuman psychology as they go.
Speaking of the psychology, this is a book clearly written by someone who read a lot of Freud. Everything, and I mean everything, comes back to sex and mothers and childhood trauma.
You can see Burdekin thinking her way through certain ideas even more than in the other two novels, and she didn't exactly hide the fact that the other two were extended thought experiments (The End of This Day's Business more than Swastika Night, but still...). For example, she spends a long time in the beginning talking about same-sex relationships as "deviant" and "perversions", but also says that the government should encourage them as a way to control the population, and later says they're quite harmless. She also acknowledges the existence of gay people in every day society a lot more than other novels I've read from this time period. I have no idea what Burdekin actually thought about gay people. I suspect maybe she didn't know either. (Daphne Patai refers to the female companion and housemate that she raised children with as her "friend" throughout the afterward of The End of This Day's Business. Since Patai interviewed her companion (though not Burdekin who was long dead by then) and presumably used this term at her request, I shall not insinuate about things that we know nothing about.)
I also suspected that she didn't know how she felt about the society she created in The End of this Day's Business. Burdekin seems to have spent a lot of time sorting out her own mind in novel form. I'm not complaining--I find her stuff really interesting--but I am rather surprised that she managed to get it published. Maybe that's my own ignorance of other experimental literture talking, though.
So yeah. Not exactly a book I can recommend in the 'go out and read this you'll love it' sort of way, but not something I'm sorry to have read either.
(Now I want to read Quiet Ways by the same author, but when I looked it up on World Cat, the only libraries that had it were in the UK. Somehow I doubt ILL covers international shipping, and even if they did I'd feel guilty asking them to.)
[1] I have nothing against airport novels. They are a perfectly respectable way to pass the time. They simply don't tend toward the abstruse and philosophical as a rule is all I'm saying.