So, I hunted around to find the customer care email (they do hide that thing, don't they?). And they got back to me fairly quickly. What I discovered is that I must have put passwords on some pages (I suppose that's what happens one one starts building a site 3 months ago and then quits...one forgets what one has done.) Which also happened to be the home page. So I was caught in circular log-in pages, but I couldn't tell because they are identical.
So. I'm putting the "fix" here for other people to find. I was wrong so you don't have to be!