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 <i>appearance</i> of the wing being angled upwards.<p>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.<p>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.