This page confirms what d.west said. An excellent photo, (click it to enlarge,) accompanys the definition. A nice array of other cowboy terms and photos thereof can be found there, too.

Hoolihan: A style of loop used when throwing a rope: a loop thrown over the head with the wrist turned backwards often used for roping horses because the rope is not swung before it is released, so it does not excite the animals.