Can a helicopter stand on the ground upside down on its rotor head?
Stand on its rotor head? yes, maybe. Spin? not for long.

The rotor hub and blades are quite strong and can take the helicopter weight at-least temporarily (leaving aside the balancing issue). The blades are designed to overcome stresses in flapping and can take the load. The rotor hub however, is not designed to take loads in this attitude (it is designed to lift the helicopter, and not push it) and its capability is an open question. You could put the helicopter upside down, but its going to be a brief affair with no guarantee that the parts will ever work again properly.

However, in case you want to spin the helicopter, you run into a number of issues. You could clamp the blades down and try to spin, but the rotor hub will struggle to take the load. The load path from the rotors to the fuselage is designed to take lifting loads and not this. 

Helicopters are not designed for sustained inverted flight and as a result, the systems are not designed for anything like this. For example, the gearbox lubrication system isn't going to work for long in the upside down condition. The rotating control rods aren't going to take the gear box weight, let alone the helicopters'.

But first, the dynamic loads- helicopter systems below the rotor hub are not designed to take rotating loads, let alone the horribly unbalanced one like a fuselage. Even the lightweight helicopter blades are balanced to prevent and unsafe vibrations -in a case like this, the rotating fuselage would most likely tear apart the helicopter before anything else.