The unit powers up. Before the DTV switch (yes, that's how
long it has been neglected, here), it would receive analog
broadcasts but display them "offscreen". I.e., imagine dragging
the image (the entire rectangle that surrounds it) up and to the
left so that only the right side and bottom remain visible on the
physical screen. (!)

Two possibilities. Check to see where the "offset" is set (the controls that
let you position the image on the screen). Also check to see where the
"display" setting is -- automatic, dot-by-dot, full screen, 4:3, etc.