How about on the other way round?
To post updates from Facebook to Twitter automatically when you post your updates on Facebook.
Yes, you can do this by login to your Facebook and then visit
Over there, you can link your all the Facebook Pages you have to your Twitter account. So, every time as you post your Status Updates, Photos, Video, Links, Notes and Events in your Facebook page, it will automatically publish to your Twitter account too. Hence, this is so convenience and save your time!