The primary cause of cyclones rotating counter-clockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere is the Coriolis Effect.
The Coriolis Effect is a result of the Earth's rotation. In the Northern Hemisphere, the rotation causes moving air (and fluids) to be deflected to the right, resulting in counter-clockwise rotation around low-pressure systems. Conversely, in the Southern Hemisphere, the deflection is to the left, which leads to a clockwise rotation around low-pressure systems.
While the jet stream, prevailing winds, and low-pressure air moving upwards play roles in the formation and behavior of cyclones, the specific direction of rotation is directly due to the Coriolis Effect.