Curiouser and curiouser.
I ran an old copy of a file manager, ZTREEWIN, which amongst other things lists all files at a given location (hidden or not). I did a full disk search and, surprise, surprise, it showed a NORMAL.DOTM, with location the template folder. I exited that program, tried the command prompt DIR /AH again on the template directory, and this time the .DOTM file showed up.
Reran Word, which now shows the .DOTM file at the template location.
Problem apparently solved, but clueless as to why.
Thanks again for your help.