Good question. I recently reinstalled Thunderbird at home, and it's connection manager had an auto setup for Gmail connections. Perhaps that would be a good place to look atm. (I am at work atm, otherwise I would check for ya). I know it doesn't look at labels properly (which are 100x better than folders imo.) Curious though, it may just have more compatibility to Gmail than I thought.

As for invitations, do you mean to gmail? Cause it is public now.