Because a membership type was in use, I wasn't able to edit its relationship properties via the CiviCRM GUI. So I did it in MySQL (by editing membership_type_id and membership_direction fields in civicrm_membership_type).
I thought I created all related records the civicrm_membership table (each contact needs their own, even if the membership is inherited), but I must have missed some or encountered a bug because after doing this edit, I had to go through and Edit then Save some membership records in CiviCRM itself to fix them.
However this didn't work for everybody, and records with multiple relationships only passed on memberships to one of the two related records. Again, this may have been the result of an oversight on my part.