The rye bread sandwich is named for its bread because the bread is the part that changes everything else. Rye is dense, close-grained, and faintly sour, with a chew that resists the teeth and a flavour with an edge to it rather than the neutral sweetness of soft white. In a British catalogue that runs mostly on yielding floured rolls, rye is the loaf you reach for when you want the bread to argue back: to have its own taste, hold its shape under a wet or assertive filling, and stay structural through the afternoon instead of collapsing into the meat. The crumb is not a background carrier here. It is half the flavour and the reason the sandwich is built the way it is.
The craft is matching a strong bread to a strong filling and cutting it thin. Rye is heavy, so a thick slice reads as mostly bread and mutes whatever is in it; cut thin it has presence without burying the filling, which is why a rye sandwich is almost always built on slices, not a roll. Its sour note and firm crumb suit fillings that can stand up to it and would be lost in plain white: smoked fish, salt beef, pastrami, a sharp cheese, a strongly cured meat. The dense structure is also a moisture advantage, since rye absorbs a wet filling more slowly than an open crumb and stays intact longer, which is why it carries pickled and brined things well. Butter or a sharp mustard goes to the edges to bridge the bread's edge to the salt of the filling and to slow what soaking there is.
The variations are mostly a question of the rye itself and what it is asked to hold. A light rye is milder and closer to white; a dark, dense, almost black rye pushes the sour and the chew to the front; pumpernickel takes that to its extreme as a near-cake slab for the strongest fillings. The smoked-fish and the salt-beef and pastrami builds are the fillings rye most often carries. Each deserves its own article rather than being crowded in here.