Well, despite Azure offering Linux in their cloud, the Linux support for signing up is lacking. If you’re on Linux (Firefox or Chromium) you get the following screen which is stuck in an endless-loop:
Turns out, you have to be on Windows, in order to sign up through your browser (Chrome worked just fine). The bit where it was hanging on Linux was the credit-card check.
Oh dear, they still have a long way to go…