Thanks – turns out it was Jetpack! It was also screwing up the slideshow on the home page.
OK, so turning off Jetpack (malignant evil – heh) actually did nothing.
What seemed to be conflicting was Social, the Mailchimp Social Broadcasting widget. Turning it off fixed everything.
Thanks for the reminder – when in doubt, assume it’s a plugin!
No, it’s not fixed – that example page shows that if people comment on blog articles on Facebook, the comments show up on the blog, but there’s no standard, stand alone, wordpress comment form showing up on the pages.