After the clean Guest profile pretty much ruled out internal issues I went to the site's tech support forum and sure enough there was a whole thread of complaints about sessions being terminated on logout.
The problem was finally resolved today.
Opera has so many quirky little problems it's easy to suspect it as the problem causer.
On the positive side, using the pristine profiles in my Windows Guest account is great new tool to help isolate issues with both Opera and Chrome.
Thanks again for the help.