In the Page Editor, I don't see it and can't click on, but if I switch to the Page Structure outline view, I see it under Template Header where it belongs, and if I click on it, I see an empty outline in the Template Header preview where it should be; I can also see its properties in the Property Editor inspector on the right. I've compared these settings to another Napili community where the Profile Header is present, and I can find no differences. I've checked every community setting I can think of. I've also confirmed that I'm using the latest version of Napili Summer '16.
Also possibly relevant - the search box is also present in the Page Structure outline, but not visible. I don't want it be visible, but I don't know how/why it is hidden or if it's related.
How can I make the Property Header visible again?