So. let's discuss dog box transmissions. No, not dogleg transmissions — that name refers to the shift pattern, while "dog box" describes internal gears, though a transmission can be both at once. Dogleg describes the shape of the shifter path from first to second (which isn't the weirdest shifter or gear selector in the world, but weird enough that it tripped up James May in a (likely faked) bit on "Top Gear" where he kept shifting into reverse by accident). But a dog box gets its name from the transmission's internals. A dog box transmission works thanks to the "dog rings" or "dog clutches" inside that feature teeth resembling those in our canine pals. These teeth are large, brutish, and capable of handling fast engagements.
The reason you find dog boxes in performance cars and racing cars is that the engagement is anything but smooth. In a normal synchronized manual transmission, friction is what lets the gears mesh, meaning that the gearbox can give some grace to a driver with bad technique by allowing some slippage. There's no such grace in a dog box. In a dog box, you'll see shifter forks as you would in a normal synchro transmission, but instead of being connected to synchronizers, they are connected to rings with these dog teeth. Shift slowly and you won't engage the teeth with the gears. Move fast and you'll get the lightning-quick shift you want.
Why have such a brutish transmission? Without synchros, the gears are going to wear down and shifts will be harsh, right? Well, in racing scenarios, speed is everything, so wear is just part of the package. Consistent power delivery is what really matters. Besides, dog boxes are designed for the abuse thanks to overbuilt, durable gears. No, those gears aren't necessarily straight cut, though they can be. You'll find both helical and straight dog ring teeth.
What's really cool is, while a regular non-synchro transmission requires double-clutching, with a dog box, you don't even need a clutch when shifting. You can bang away shifts as fast as you want. In fact, go ahead and preload the shifter by pressing it up against the next gear when downshifting. When you're ready to shift, blip the throttle and push it in when you feel the gear speeds align. You only use a clutch with a dog box to get the car moving, kind of like an F1 car. After that, slam that lever into each gear so fast that spectators will think you've got an automatic.
Let's do something that's way better than jabberjawing about how dog boxes work. Watching someone demonstrate a dog box transmission will give you an idea of how quick you need to shift. JPM Performance has an excellent demonstration using a Miata sitting on a dyno. You can see just how quickly the shifts happen, which is certainly aided by the ludicrously short throws on the shifter. Plus, no clutch action! The only thing needed is a momentary release of the throttle for upshifts and a little blip for downshifts.
Look, dog boxes and dogleg transmissions are both awesome. Look at the Pagani Utopia and its dogleg manual transmission. That thing is to die for. Doglegs are super common compared to dog boxes, though. While you'll find dogleg transmissions in the aforementioned Pagani, Mercedes 190E 2.3-16, M30 BMW M3, Lamborghini Countach, and scores of others, the only car to get a factory dog box was the Abarth 695 Biposto, and even then, it was optional. You can watch an Abarth 695 with a dog box in operation on the Seen Through Glass YouTube channel, and the shifter looks glorious.