I feel like I am missing something obvious that I just can't quite put my finger on.
As an administrator, I see "Add Content > Book page" and "My Workbench > Create Content > Book page" in the admin menu, and I see tabs to create and edit drafts when viewing a book page.
My "book author" role does not see any of this though, despite seeing controls for 3 other content types just fine. The node permissions allow book authors to "Create new content" for the book page type. The 3 other content types also have "Create new content" and no edit permissions, and they all work fine.
For the 3 working content types, if I remove the "Create new content" permission, the admin options disappear from the menu and tabs aren't shown, and the menu and tabs reappear as soon as I recheck "Create new content". So it's just this Book module content type that's giving me problems.
This is a Drupal 7 site, with a few contributed modules that affect content access, such as Workbench Moderation and Organic Groups. I have also tested out some other content access modules in the past, such as Workbench Access and Domain Access. All these modules have long been disabled, however, and I have rebuilt permissions several times since. I suppose it should be noted though that I only just recently enabled the Book module, while the 3 working content types were all in use when I was experimenting with Domain Access, Workbench Access, et al, so maybe there are some lingering settings in the database?