PilotJames,
picture the aircraft remaining in upright level flight and performing a bunt. You'd push the stick forward, first unloading the airframe (zero g). At this point, the angle of attack has been reduced to the where the wing is not generating any lift. The stagnation point, i e the point on the leading edge where the airflow splits, has moved up on the leading edge.
Push the stick further forward and the angle of attack reduces even more, and the lift with it... but wait a minute? You were already at zero lift? Aha! Negative lift indeed!
Now, keep pushing the stick forward and lift continues to decrease, i e the negative lift increases. As you reach an amount of negative lift equal to the weight of the aircraft, you can flip it around 180 degrees and remain in level flight rather than bunt the aircraft.
There's a limit, of course, where you will instead suffer a negative G stall, where the airflow separates from the lower surface of the wing, similar to the separation from the upper surface in positive G stall. If that happens before the the negative lift equals the weight, the aircraft is unable to sustain inverted flight at that airspeed. If increasing the airspeed to a limiting speed does not solve this problem, the aircraft is incapable of inverted flight, at least at the current gross weight.
You can also run into the problem of the elevator authority being insufficient to lower the angle of attack enough for the negative lift to equal weight, in which the aircraft is again unable of inverted flight at the current weight.
Look at the links in my previous post and you will see that most profiles (ancient ones, WWI era, and the Clark Y are the obvious exceptions I can think of) have a convex lower surface, just less so than the upper surface.