offlineimap allows synchronizing IMAP and/or Maildir mailboxes both ways (e.g. not just a copy). Even though the name and introduction on its website doesn't necessarily carry this very well, it can synchronize 2 IMAP mailboxes, none have to be Maildir.
It also supports incremental and two-ways sync, which means you can keep using both mailboxes and it'll do its magic.

This came in very handy in synchronizing mailboxes between providers: just set it up to use IMAP on both ends, and let it do its magic -- it's fairly long for large mailboxes because IMAP isn't the fastest. And as it's incremental, you can both interrupt the sync, and re-run it as many times as you want if the source mailbox changed (received new messages, or removed any).

It also has a lot of options, including custom folder filtering or transformation and many others. For those who care, it apparently has specialized GMail support as well.