A cutting apparatus for cutting a cap provided with a cup body and having a base wall, a side wall and a protrusion projecting from the side wall of the cap outside the cap, includes: an advancement path travelable by the cap along an advancement direction, a spindle movable along the advancement path and rotatable around an axis thereof, the spindle being shaped to interact with the cap and to rotate the cap (10; 110; 210) around the axis, a support movable along the advancement path together with the spindle, said support being shaped to interact with the cap and to retain the cap against the spindle, a cutting device arranged for making a cut in the side wall of the cap when the spindle reaches a cutting zone provided in the advancement path, in which the spindle includes a hollow housing shaped to house the protrusion so as to stop in use a rotation of the cap in a preset position with respect to the spindle so as to orient the cap with respect to the cutting device.