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.
If planes are not supported by the air beneath them pushing up why don't they fall out of the sky?