Users are getting spontaneously logged out
OK, I think I've got this figured out:

As part of the siteI need to be able to programmatically set user field values -- what used to be called "profile values" in Drupal 6. I looked around for some ways of doing that, and found a possible solution at http://drupal.org/node/1047600#comment-4094988, which recommends something like this:

$existing = user_load($uid);
$edit = (array) $existing; 
$edit['field_somefield']['und'][0]['value'] = 'blah';
user_save($existing, $edit);


The instrumentation I did showed user_save was the guy calling _password_crypt(). My belief is that, somewhere along the way (either the method itself or the way I was using it), my call to user_save() was trashing/regenerating user->pass, leaving me in my unfortunate state.

So: I'm now doing the field-saving stuff in a way that doesn't involve user_save (some day I'll figure out this new entities stuff, but, in the meantime, I'm just bashing values into the database) and, so far, I'm staying logged in.  Hopefully this will solve the problem; thanks to all for your attention.