Turns out it's not as bad as that! I wasn't emailed my password. I was emailed a link.
http://appcenter.html5tools-software.intel.com/csd/ForgotPas...
That's fine. So there's this unique token that means only I can reset my password... Except... Let me decode that token...
EYl0htUzhivYzcIo+zrIyEFQUE1PQkk= -> t3+(:APPMOBI
So does this mean I could reset anyone's password, given their email address?
Anyway I went to the page and put a new password in (trying to follow the unstated rules found previously). When I pressed submit, the page darkened and I was told 'Error 500'.
Tried again with a new password and was told 'Password must contain one of the following !@#$% special character[s].[sic]
Well, I was told it for a second, but the message went away really quickly, so I had to keep submitting the password it didn't like until I'd remembered what the characters it required were.
Anyway the password reset link doesn't appear to expire, so if you fancy logging in to Intel AppCenter as me, just reset my password. I can always reset it back if I want to use it.