The device comprises a path for the web material (N) and, along said path, at least one cutting unit (7A). The cutting unit comprises: a plurality of intermediate disk-shaped blades (9A) disposed on one side of the feed path of the material, which can be positioned transversely to the direction of feed and cooperating with a fixed counter-blade (33A) disposed on the opposite side of said path. Also provided are two lateral disk-shaped blades (10A) to cut the trimmings; and a member for transverse positioning of the blades. The two lateral blades each cooperate with a respective movable counter-blade (71A), separate from the fixed counter-blade, which can be positioned transversely to the direction of feed of the web material.