Following up, again, headers and footers are the key.
These are properties of sections. Every section has three headers and three footers even if you never see them. A page can have multiple sections; a section can have multiple pages.
Header/Footer Settings Recap

You can have a first-page (of the section) header, an odd-page header, and an even-page header. The same is true of footers. Page numbers, to be shown only on even-numbered pages, go in the even-page header or footer.



Each kind of header/footer can be independently "linked" to the header/footer of the same kind in the preceding section, or not.

Thanks for the link. I think I figured it out. I had 4 sections, all of them said they were linked to the section before. I disconnected all the sections and renumbered them from scratch. It looks right but to be sure I'll have to convert it to PDF, send it to the publisher, and have a copy printed. Fingers crossed.