How to convert an existing Azure Web Role to a new-style Azure Web Site?

Is it possible to convert an existing Azure Web Role (which is ASP.NET MVC) to a new Azure Web Site? Other than just creating a new project and copying the code over to the new project so that it looks like the old application, that is.

Yes, you are right. Windows Azure Web-Sites are shared sites with others on the same machien and you don't have capability to Remote into. You sure can manage the sites direly from Git or Powershell. Also new Azure web-site design does not use any of the Web Role Model so you just need to take your ASP.NET website application and export as your new ASP.NET Web site.