I checked the settings in IE and found that, as I had expected, cookies ARE enabled for all sites and that should not be the problem.
I have done some more testing and found that if I go to the perlmonks.org site then all of the links across the top and any hyperlink on any web page causes me to be "logged off" when I get to the next page.
If I go to perlmonks.com, then I do not have this problem. I stay logged in as expected.
Sounds to me like IE is configured to accept cookies from perlmonks.com but is configured to reject cookies from perlmonks.org. It can be quite difficult to determine why that is the case.