I\'m experimenting with the Microsoft Azure portal in order to see how my legacy application performs with the least amount of rewrite. Authentication is a problem.
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Thanks to Simon W above, I found my problem. In the Azure Portal, there is a setting that controls "Azure Websites Authentication / Authorization". Its found under Websites, then Configure, then scroll down a bit to "Azure Websites Authentication / Authorization". In my case there was an entry there: my website was tied into the "Directory" for my account and the "application" was set to my "tenant" (I think that's the Azure term).
To fix this, I simply removed the entry in the portal and I am able to login to my web application perfectly. Forms authentication wired up to Membership is working as I'd expect.
I believe that I published the original website using VS.NET 2013 Publish (using Server Explorer with the Azure SDK tools installed) and maybe there is a default setting that enables Azure AD in that scenario. I won't know until I retrace my steps and confirm this later. When I create a new website using the Portal, it does not default to this scheme.
Simon, I don't know how to give you credit for the answer... you basically got my head pointed in the right direction (just the use of "Azure AD authentication" got be googling the right terms) and facilitated the fix.