Spinning on the swing

 We wound up the swing and then it span when released.

Hooking the chains over each other pulls them in and ensures that there is a horizontal component to the tension force in the chain. There is a horizontal component in each of the pair of chains pulling in opposite directions. The key is that the way the chains are wrapped side by side, so the horizontal components are not perfectly lined up. This produces a couple, a turning moment. The swing rotates.