The Quaife differential works on an entirely different principal. Each axle is attached to a gear, called the sun gear, and each sun gear has several small gears around it called planet gears. The planet gears from one side mesh with the planet gears from the other side, so when you put both drive wheels in the air and turn the left wheel clockwise, the left planet gears turn counterclockwise. They turn the right planet gears clockwise, which turn the right wheel counterclockwise. In other words, the wheels spin in opposite directions just like a normal open differential.

The magic comes from all the gears having their teeth cut helically. That means the teeth are angled so that if the gear were long like a log, the gears would spiral around the log in a helix. When you put a big load on a pair of helically cut gears, there's a reaction force pushing them sideways.

Look directly at the interface between two gear teeth and the reason for this sideways force is pretty obvious. The gear teeth meet at an angle, and when you try to push straight through an angled interface like that, you get shoved sideways. The latch on the front door of your house works the same way. Close the door until the latch touches the striker plate and you have a similarly angled interface. Push straight on the door, and the sideways reaction force pushes the latch into the door. In a helical differential like the Quaife, the reaction force shoves the sun and planet gears sideways, where they rub against the ends of their housings. This rubbing encourages both wheels to turn together.

The reaction force only happens if there's some resistance to your shove, though. If there was no resistance in our door latch example, the doorframe would just fall over. If there's no resistance from the other gear, it will just turn. This is why, technically, a helical limited slip isn't really a limited slip. Put one wheel on ice and the other on pavement, and there won't be enough resistance to prevent all the power from going to the ice.