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.

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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.