Well, I got this fixed. The problem was way too much cable slack on the lowest gear. So the first shift was mostly taking up the slack, not shifting the derailleur enough for a true shift. I took up all the slack and now a single full shift in either direction moves the derailleur sufficiently to effect the shift. (And a half shift will trim the derailleur without shifting.)
(Somewhere there is a planet with every bike perfectly adjusted, populated by creatures with three hands!)