If you take an aerobatic aircraft with a non-symmetrical wing section (a Bucker Jungmeister would be nice, or CAP 10, or Zlin 526), roll it upside down and trim for inverted flight, the aircraft will be just as stable as right-side up. Viewed from the outside, however, the wing will now be producing a nose-up pitching moment and the tailplane will probably need to provide an up-load to trim.
A canard like the Rutan LongEz is an example where the 'tailplane' produces an upload in the 1g trim condition and the aircraft is stable.