OK, I called the number on the Customer Care link and explained my problem.
I was told that was a technical problem, then I was transferred to "technical support".
I explained the problem to the guy there and he asked for a "Customer Number" or "Support ID". I explained I didn't have either of those - I'm not an existing Symantec customer and I don't have a Support ID (presumably, because I can't log into MySupport).
He said this was a problem and said he'd have to transfer me back to Customer Care to get one of the above. I got disconnected at that point and don't have time to call back at the moment.
I have reset my password on my normal account to see if that did anything, but to no avail.
If time permits later I will try creating a new account.
edit: actually I just tried creating a new account in an entirely different browser. Exact same problem.
For clarity, I have tried this process in Firefox (v6 and v7), Chrome, and Internet Explorer 8. I am running Windows XP.