In Looking Backward, how does Edward Bellamy reimagine the role of women?

For a novel with radical ideas given the time in which it was published, Bellamy speaks only briefly on the role of women in his "new America," instead dedicating most of the book to discussion of economic changes. The most prominent feature of the role of women is the legal power women have over their own affairs. In the book, women have their own female high-ranking government official with ultimate say over how women are treated, are tried by female judges, and work in a different general labor system meant to compensate for less physical strength while giving equal compensation. Significantly, when these aspect are combined it removes any financial or social dependence on men, allowing women to choose lifestyles and partners for themselves. For a nineteenth-century reader, this idea of full rights and self-control would have been shocking. In the later chapters when Bellamy eventually discusses women's rights in his utopia, it becomes clear that there is an idea of separate but equal. While controversial today, this notion would have given women far more leverage than was present at the time the novel was published.
Bellamy remained a product of his time, leaving the domestic work of childcare to women; however, he made it clear this was considered an equal type of work. It is likely that the concept of true gender equality was either beyond him (those ideas did not become prevalent for many decades) or was deemed too radical for his predominantly educated, male readers to accept.
Bellamy discussed the role of women in much greater detail in his second book, Equality.