Unit (4) for molding preforms (2) that each have a body (15) and a neck (17) extending the body (15), this unit (4) including: a frame (19), —a mold holder (27) secured to the frame (19) and carrying at least one mold body (29) having the impression of the body (15) of the preform (2); mobile equipment (60) including a carriage (61) and a core holder (62) carrying at least one primary core (64A) and one secondary core (64B), the carriage (61) being mounted so as to be movable with respect to the frame (19) between a molding position and a demolding position, this core holder (62) being rotatable with respect to the carriage (61), in the demolding position of the latter, about a rotation axis parallel to the molding axis in order to transpose the cores (64A, 64B).