I have seen lots of posts about "NO sender, NO Subject", but one of my Office 365 accounts is: "No Sender" but everything else is fine.

Same issue on two different 8.1 machines.

What on earth has happened?

Hi

It seems to be a known issue. Are you using mail app on Windows 8.1? try moving your cursor to the right hand of the screen while in the mail app

Choose Settings->Options

Turn off Group conversations and then try re-syncing your office 365 account on Mail app, then check. If the issue persists, please reinstall mail app, be sure that your mail, appointments, etc. is backed up.

https://social.technet.microsoft.com/Forums/en-US/b1293488-f853-45d6-84cb-8b0e04ad4f8d/windows-8-mail-app-no-sender-no-subject?forum=w8itprogeneral

Since you mentioned that this issue happened on two machines, Id suggest that we also check with Office 365 community forum see if it is a known issue. There you can get more effective suggestion by other experts who familiar with design of Office 365.

Regards,

D. Wu