In a large pot, cover the potatoes with water, add a large pinch of salt and bring to a boil. Cook over moderately high heat until tender, about 20 minutes. Drain the potatoes and return them to the pot. Shake the pot over moderately high heat for about 10 seconds to dry the potatoes. Transfer the potatoes to a large rimmed baking sheet and let cool completely. Peel the potatoes and cut into 1-inch pieces.
In a large bowl, blend the mayonnaise with the relish, vinegar, mustard powder, paprika, black pepper, cayenne, radishes, celery and onion. Gently fold in the potatoes and season with salt. Refrigerate the potato salad for at least 4 hours or overnight. Serve cold or lightly chilled.