If planes are not supported by the air beneath them pushing up why don't they fall out of the sky?
IOW a classic airfoil that appears horizontal is really best thought of as a flat surface that's angled upwards. When a plane with a wing like that inverts, it really does "lift" downwards, and it takes considerable elevator correction to keep level.
If the wing had a symmetric airfoil section, then it needs to be visibly angled upwards in order to fly; but on the other hand when upside down it doesn't take as much elevator input to stay up as the classic asymmetric version.