Bring a large pot of lightly salted water to a boil. Cook fusilli in the boiling water, stirring occasionally, until tender yet firm to the bite, about 12 minutes. Drain.
Combine basil, Parmesan cheese, pine nuts, oil, and garlic in a food processor; blend to make pesto.
Toss pasta with pesto and mix in red onion, radicchio, and mayonnaise. Season with salt and pepper. Refrigerate for at least 30 minutes before serving.
