The cloud is suspended in a fluid that is also spinning. The atmosphere is an ocean we live at the bottom of, and where it makes contact with the earth friction between the fluid and the solid conveys inertia from the solid to the fluid. Inertia is likewise conveyed, by friction, up the viscous column of atmosphere.

It is the same principle by which stirring your coffee causes the entire contents of the cup to swirl, and not just that touched immediately by the surface of the spoon.