A few modern helicopters can perform a roll and are hence flying upside down for a few moments but they cannot maintain sustained inverted flight, unlike a fixed wing aircraft. Older machines did not have the power or the rotor technology to make rolls a safe aerobatic option.

The short answer is yes, but with major caveats. Most standard commercial and military helicopters are not designed for sustained inverted flight because their lubrication and fuel systems rely on gravity; flipping them would cause the engine to stall. However, specialized aerobatic helicopters, like the Red Bull BO-105, feature a rigid rotor system that allows them to perform loops and rolls. In a rigid system, the blades don't "flap" as much, preventing them from striking the fuselage when the aircraft is inverted. Even for these specialized birds, flying upside down is usually a momentary maneuver rather than a sustained cruise. If a standard helicopter were to attempt this, the negative G-forces would likely cause the rotor blades to flex dangerously or the mast to snap under the inverted load.