Stir the mayonnaise, ketchup and relish together to make the Russian dressing.
Spread dressing on one side of each rye slice and build each sandwich with Swiss, corned beef, sauerkraut and another Swiss slice.
Close with the remaining bread and butter both exterior sides.
Cook the Reubens in a covered skillet over medium-low heat for 4 minutes until the first sides are deeply browned.
Turn, cover and cook for 4 to 5 minutes more until the cheese melts and the corned beef reaches 165°F.
Rest the sandwiches for 2 minutes and cut them diagonally.