You can have Statuspage automatically tweet updates when you create or update an incident.
To do this, you'll need to activate the Twitter add-on in the add-on store first. Once you do that, you'll see a link to the Tweet Updates page added to the left navigation.
Statuspage.io automatically prepends the update with your designated Tweet Prepender and the current incident stage (Investigating | Identified | Monitoring | Resolved)
Here's what this looks like on the Twitter side.