A method for orientating a focused laser cutting beam eccentrically relative to the nozzle axis of a cutting gas nozzle and a laser processing machine for performing the same, the method including: arranging a redirecting mirror in a beam path of the laser cutting beam upstream of the cutting gas nozzle, the mirror being rotated about a direction which is coaxial with the nozzle axis and/or about a direction which is perpendicular relative to the nozzle axis and which corresponds to the beam incidence direction of the laser beam on the redirecting mirror. The cutting gas nozzle and the redirecting mirror are arranged in a second structural unit of a laser processing head, for rotation about the direction corresponding to the beam incidence direction relative to a first structural unit.