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 Apps section. Once you do that, you can Edit your Twitter settings from the Apps section at any time.
Navigate to the Tweet Updates tab in the left navigation and click Authenticate With Twitter.
Once you authenticate your Twitter account, you can configure a few settings, including...
Note: Custom Twitter card images will not show up if your page is in Trial mode.
Statuspage 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.