We'll explain a new proof of FOOO's surgery formula and an extension to the case of clean intersections. This leads to an understanding of the connecting map of projective twists and a complete solution of Huybrechts-Thomas's conjectural formula for projective twists. The main tool is to formulate an immersed Lagrangian cobordism theory, which also leads to insights on other basic issues on immersed Lagrangian Floer theory.