Thanks for your response, pbcopter. Sorry that my post was not clearer.
I run two computers - a desktop and a laptop - on the same home network. The desktop - which is new and just had Office 2010 installed - has the problem described. The laptop - which is old and has XP Pro 2002 installed - does not have a problem. Hence, despite the desktop having a problem, I can use the laptop to check independently for emails to all 4 addresses, which is what I was doing when I said that my tests worked for each acccount.
All this said, some good news - my problem, i am embarrassed to say - got solved about an hour ago when, while repeating for the umpteenth time the drill of test-clicking everything I could in Oulook to see what it opens and where that might lead, I keft-clicked (accurately, I guess, for the first time) one of the tiny white arrows in the navigation pane to the left of the non-default accounts. Voila! (As those experienced with W2010 (and Vista?)) will know) it turned black, the invisible subfolders all appeared and my queation was answered.
I had been looking very hard and very long, but in the wrong place, for this oh-so-obvious fix. Windows 2010 must have got me disoriented, I guess...