This problem is now fixed, due to help from Chris H. in
the TabletPC newsgroup. Thanks, Chris. Here's what I
learned/did.
The problem was that a third-party application (for a
smartcard reader that I didn't need anyway) had replaced
the standard Windows logon file (usually msgina.dll) with
something else that was interfering with Remote Desktop.
Chris suggested this KB article -
http://support.microsoft.com/default.aspx?scid=kb;en-
us;300470&Product=winxp
and it provided the path to the solution. Uninstalling
the offending software did not fix the problem by itself
though, so I had to edit the registry (careful out
there!) to replace the default Gina setting in Winlogin
with msgina.dll, and now it all works just fine.