Bingo -- that's what I forgot to do, set the message type to HTML instead of plain text (the default).

I was misled by the fact that Hotmail seems to do this automatically if there's an HTML tag at the start of the document. So I'd paste the same message into Hotmail and send it, and that was being delivered as a type-HTML message, but when I did this programmatically it was still delivered as plain text. (Text that just happened to be a valid HTML document in this case.)

Thanks for your help - Doug