[injb]

There's a very famous book called Stick and Rudder that does a good job of explaining this. A plane stays up because of Newtonian reaction. The Bernloulli principle and its associated airfoil shape should be thought of as an efficient way of displacing air downwards without the appearance of the wing being angled upwards.

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.