Combine the sweet-and-sour sauce ingredients in a small glass or enamel-lined saucepan. Cook over moderate heat, stirring, to dissolve the salt and sugar. Let the sauce cool.
Peel the radish, then slice it into thin (1/16-inch) rounds. Lay the rounds out on a cutting board and sprinkle them with the salt. With your fingertips, gently rub the salt into the radish slices. Let the slices sit for a few moments, then rinse them in cold water. Pat the slices dry, then place them in the cooked sauce. Let the slices marinate for at least 15 minutes at room temperature, or up to 24 hours covered and refrigerated.
Lift the radish slices from the sauce and pat away any excess liquid. Lay a single slice on your cutting board. Lay a slice of smoked salmon on top to cover it; cut and trim the salmon as necessary to achieve an even layer. Top with another slice of radish; gently press the “sandwich” together. Repeat, making as many sandwiches as you can from the radish and salmon slices. Line up the sandwiches on your cutting board, cover them with clear plastic wrap, and place another cutting board on top to weigh down the sandwiches. Refrigerate for at least 10 minutes and up to 12 hours.
Remove the top cutting board and slice each sandwich in half. Arrange three or four of these, leaning against each other domino-style, to make a single portion.
Rinse the radish sprouts under cold water, shake off excess moisture, and trim away the stems. Or rinse and trim the sprigs of watercress. Garnish each serving of Red and White Half-Moons with some radish sprouts or a sprig of watercress.