I know planes can manage it, but can a helicopter?
Some can, some can’t.
The old UH-1 that I grew up with could not. The rotors were attached to the hub in such a way that if you turned it upside down Bad Things would happen. The UH-60 Blackhawk can and does fly just fine upside down.
It depends on what you mean by “upside down”. I don’t know that it would be physically possible for a helicopter to fly upside down with negative g, that is, with the force of gravity acting toward the top of the rotor system. However, there are a number of helicopters that can perform loops and rolls and such manoeuvres that included inverted flight but the g forces are maintained toward the bottom of the helicopter due to centrifugal forces.