It's not possible for the TD to directly post to your twitter feed, but that doesn't mean it's impossible altogether. Actually, one could modify the statusListener.php script to post to a twitter feed when it receives an updated status (or every N status updates, or only once every N minutes, for example). The TD is providing the information. Really anything could happen to that information after the TD hands it off.
Of course, that relies on someone coding that up. I'm not a twitterer, so I'm definitely not familiar with their API. I imagine it would be fairly trivial for someone familiar with their API.