I am posting this as a question and answer for anyone else that has this issue as it was an absolute PITA to debug.
We have a taskpane add-in for Word that has a login page using cookies to store the session state. This works completely fine in everything except Word Web in Internet Explorer 11 on Windows 7.
By using Fiddler we can see that our cookies were getting set and everything is fine, even the session cookie stays the same (this is another issue altogether), but for whatever reason the .AspNet.ApplicationCookie is never getting sent back with the request for the HTML page - but it is getting sent back for the request for the JavaScript. WTF.