I spotted that "GoogleUpdater is trying to access the Internet" from ZoneAlarm.
I'd removed Chrome (didn't really "do it" for me) - but the "updater" was still there and trying to do something. Strangely, although "Search" could find the GoogleUpdater.exe file, Explorer - with "show hidden files" turned on - couldn't see it, or even the folder where it lurked.
It was still present in msconfig and in the registry, too.
I did the full "manual virus removal" process on it - which took quite a time. No way is that ever coming near my PC again.