Airplanes fly because their wings are curved on the top and flat on the bottom, allowing them to harness the Bernoulli Principle. With speed, the air passing over the wing is less dense than the air under it, creating lift. Helicopters have rotor blades that also create lift. As the blades spin they move the air. Less dense air passes over the blades with denser air passing underneath it.
Put that way it sounds simple, but it's far more complicated, because while airplanes get lift naturally by moving through the air helicopters have to constantly generate it themselves. This is how they can takeoff and land vertically, and also fly backwards, sideways, and hover in place. But they can't glide if something goes wrong; they aren't aerodynamic like planes. If they lose power, they fall like a rock. Helicopters fly by sheer brute force, which is really incredible, but gives them some limits.
Planes can fly upside-down. The fixed wing means that as long as air is moving over the wing, usually by tilting the nose upwards slightly, it still has lift. As a helicopter's rotor spins, the generated thrust moves upwards with the direction of lift, so flying upside-down would demand redirecting the blades so the thrust is the opposite way, supporting an upside-down vehicle.
That said, a helicopter could fly a barrel roll if it was moving fast enough, but that's a stunt you're more likely to see at an airshow than from a news helicopter over a city.
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