A cutting apparatus is provided for cutting a cap as it travels along an advancement path in an advancement direction. The cutting apparatus includes a spindle movable along the advancement path and rotatable around an axis thereof. The spindle is shaped to interact with the cap and to rotate the cap around the axis. A support moves along the advancement path together with the spindle. The support is shaped to interact with the cap and to retain the cap against the spindle. A cutting device is arranged to form a cut in a side wall of the cap when the spindle reaches a cutting zone provided in the advancement path. The spindle includes a hollow housing shaped to house a cap protrusion to stop rotation of the cap in a preset position with respect to the spindle to orient the cap with respect to the cutting device.