I have the same bamboo situation, the bamboo was there 40-50 years ago. Warning, any cutting of the bamboo only makes it spread and it is a fast grower. I do believe way back when, bamboo was sold for making outdoor furniture and such.
In Texas, any say so your neighbor would have begins at his property line, he can sink a barrier on his side but not force you to eliminate the bamboo on yours. After all, he knew the stuff was there when he bought his place. I have never noticed my bamboo leaning that much, did the fence get built right up against it?