Double declutch

From Trackpedia

Double de-clutch is a method used to make a smoother downshift with less wear. It was a custom of older cars or race cars, with dog-gearboxes that had no synchronizers. In a modern synchro-gearbox, it is only used in unique cases, like when making a J-Turn without locking up the tires.

The procedure includes declutching, shifting into neutral, re-engaging the clutch, maybe rev-up the engine by dabbing the accelerator in between ("midway throttle"), and than de-clutch again and shift into the appropriate gear.