To understand Double De-clutching, you first needs to know how a driveline works.
It’s much easier with diagrams, but I’ll have a go.
The Clutch is best considered as a sandwich, or rather a Ring Doughnut sandwich. It has three major components. The Flywheel, which is attached to the Crankshaft and rotates at Engine speed. The Pressure Plate, which is attached to the Flywheel, and therefore also rotates at Engine speed. The ‘filling’ in the sandwich is the Clutch Plate which is attached to the Primary shaft of the gearbox. When the Clutch pedal is depressed, the Clutch Plate disengages from the the Flywheel and Pressure Plate and rotates freely of them.
The Primary shaft of the gearbox drives a Lay Shaft. An independently rotating shaft which has a number of gears which, through moving the gear lever, can be made to align with other gears connected to the Output shaft. The only time that the Primary shaft is directly connected Output shaft is in Top gear, when the Lay shaft becomes redundant (but still rotates).
The Output shaft, obvious from its name, is connected to the driveline and will always be rotating at a speed relative to the final drive (Drive Axle).
On a Syncromesh gearbox, each of these cog engagements between the Layshaft and the Output shaft, is preceded by a polished bevel against which a bronze cone is pressured, a sort of mini-clutch, which either speeds up or slows down the helically cut cog accordingly.
On a ‘Crash Box’, the gears are straight cut and do not have this ‘mini-clutch’ and therefore the driver has to replicate this action manually from the driving seat.
When changing up. Disengage the clutch. Move the Gear lever to Neutral. Re-engage the Clutch, thereby reconnecting the the Crankshaft to the Primary shaft. Allow the revs to fall, which will have the effect of reducing the speed of the Primary shaft, and therefore the Lay shaft. The Lay shaft should now be rotating at a speed (roughly) consistent with the output shaft. Then disengage the clutch and select the next appropriate higher gear.
Changing down is very similar. Disengage clutch. Select Neutral. Re-engage clutch. Advance the Engine speed to speed up the rotation of the Primary shaft and the Lay shaft to accommodate the rotational speed of the output shaft. Disengage the clutch and select the lower gear.
Eaton twin-splitter. It’s demise was Noise. It utilised straight cut gears which are inherently noisier than helically cut gears.