The thing is that what really matters is the orientation of the trailing edge. And in airplanes the trailing edge points downwards. It is not always obvious because of the way wing profiles are designed but it is almost always the case. We could have it level, but that would mean the plane will need to always fly nose up in order to create lift, it would be uncomfortable, and most likely not ideal from an aerodynamic perspective.

When you are flying upside down the natural orientation of the wing goes the opposite way, so the leading edge points up, and indeed you can't fly level with the nose pointing straight ahead. You need to point the nose up in order to compensate the natural wing orientation, and then point up even more to create actual lift.

Of course, there are some more advanced considerations. Performance will generally be reduced when flying upside down, because the wings are not designed to do so. However some wings have a symmetrical profile, and in theory, they could be flown equally well in any orientation.