The Reuben is a hot griddled sandwich, and the griddle is what defines it. Corned beef, Swiss, sauerkraut, and Russian dressing are stacked on rye and then cooked on the flat-top until the Swiss melts down through the kraut and binds the structure, the dressing warms into the meat, and the rye crisps in butter on both faces. Built cold, those four components are a stack that slides apart; built hot, they fuse into one thing. The Swiss is doing structural work as much as flavor work, gluing a slippery, wet filling to the bread so it can be lifted and bitten without falling open. That melt is the entire reason the sandwich has its own name rather than reading as corned beef with extras.
It works because the components are a balanced system once heat is applied. Corned beef is brined and boiled to a clean, leaner profile and sliced against the grain so it yields rather than ropes. Sauerkraut is the acidic, fermented counter that cuts the fat of the meat and the cheese, but it carries water, so it has to be wrung out hard before it goes in or it floods the rye and the bottom slice gives way under the spatula. The build order is part of the engineering: the Swiss goes against the bread on both inner faces with the kraut and meat sealed between, so the melting cheese glues the structure and shields the rye from the wet kraut at the same time. The Russian dressing supplies sweetness and richness from inside the sealed sandwich rather than dripping out of it. The rye is a seeded, slightly sour loaf chosen to stand up to fat and acid without adding bulk, and griddling it in butter on a low, patient heat builds a crisp shell while the interior comes up to temperature. The timing problem is the same one a grilled cheese poses, with a wetter, heavier filling: rush it and the rye scorches over cold kraut, so the bread has to reach deep gold at the exact moment the Swiss is fully molten.
The variations are tightly bounded and each deserves its own article rather than being crowded in here. The Rachel keeps the method and swaps the kraut for coleslaw and often the corned beef for turkey, a sweeter, milder reading of the same griddled idea. Pastrami stands in for corned beef in many kitchens, pushing the sandwich smokier and richer. A grilled version on pumpernickel changes the bread but not the logic. Each is a single substitution on a fixed technique, which is the same human impulse that gave the Reuben its own name in the first place rather than leaving it as one more thing the deli does to rye.