A day or two later, they contacted me, asking for confirmation of some info (my phone number, I think), then another email saying they couldn't confirm some of my payment details, etc. ... and then followed a two-week-long back-&-forth with customer service, trying to get them to take my money.
After two weeks of this, they decided to cancel the order, said I needed to re-place the order from scratch ... except by then, the package I had originally ordered was no longer available. They're very sorry for the inconvenience, but fuck me.
I want to emphasize that I had a credit card and two different debit cards, all valid forms of payment, in my name, that I've used at various times to order things online. To this day, I have no idea what the problem was, as they never told me.
tl&dr: HB stops online fraud by (I guess) erring on the side of caution, and periodically alienating legitimate customers. Now I will never shop there again, and routinely warn others not to.