Hi Vasili–
The product pages just started working–I have no idea why. I wasn’t seeing 404–I was seeing a message that said something like “Well, this is embarrassing. We can’t seem to find the page you requested.”
Or words to that effect. I did open the Permalinks page, but wasn’t sure what to do with it, so I closed it without making any changes.
I’m thrilled that everything’s now hunky-dory, but I’d like to know why it wasn’t working before–any theories?