Unit (4) for moulding preforms (2) that each have a body (15) and a neck (17) extending the body (15), this unit (4) comprising: a frame (19), - a mould holder (27) secured to the frame (19) and carrying at least one mould 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 moulding position and a demoulding position, this core holder (62) being rotatable with respect to the carriage (61), in the demoulding position of the latter, about a rotation axis parallel to the moulding axis in order to transpose the cores (64A, 64B).