This is a continuation and refinement of a previous question.
On a plain power on, my Mac Mini displays the apple symbol and never anything else.
On a power on while holding down the Option and R keys, it displays the apple and a progress bar. When the progress bar completes, I get the log on dialog and can log on. It behaves completely normally through every process I've tried, even for days. But if I log off (or restart) (I don't remember at the moment if the effects on logon of logoff and restart are the same) the only way to get it open again is with Option and R at power on.
Does this suggest to anyone what the specific cause might be?