A bi-directional curvilinear transference apparatus for a glassware forming machine to be mounted on an apparatus support frame to transfer a parison held by a transferable and open-able neck ring mold from an indexed unidirectional rotary inverting apparatus at an intermediate station to a blow forming station, comprising: mounting means mounted on the apparatus support frame; a driving mechanism mounted on said mounting means; an oscillatory hollow arm mounted on said driving mechanism to provide it an oscillatory movement; a parison transference mechanism mounted on said oscillatory hollow arm, including sliding carrier holders (327, 327') mounted on said oscillatory hollow arm, including holding arms normally maintained closed, and gripping fingers coupled to the holding arms for internally gripping the transferable and open-able neck ring mold; and a two-positions opening mechanism coupled to the carrier holders (327, 327'), so that, at a first position of the two-positions opening mechanism, the gripping fingers are partially opened in order to grip the transferable and open-able neck ring mold gripping a parison, while simultaneously an open-able horizontal neck ring holding arms of the indexed unidirectional rotary inverting apparatus are opened releasing said transferable and open-able neck ring mold, and be retained by the gripping fingers of this transference apparatus at the intermediate station, and at a second position, the holding arms are additionally opened, in order to additionally open the gripping fingers, opening the transferable and open-able neck ring mold releasing the parison to vertically fall flat into a blow mold apparatus, but keeping the transferable and open-able neck ring mold held by the gripping fingers; and in the inverse way, said gripping fingers of the transference apparatus return the empty transferable and open-able neck ring mold closed at the intermediate station, and are further closed, releasing the empty transferable and open-able neck ring mold at the open-able horizontal neck ring holding arms of the indexed unidirectional rotary inverting apparatus to be held thereby for a following forming cycle. mounting means mounted on the apparatus support frame; a driving mechanism mounted on said mounting means; an oscillatory hollow arm mounted on said driving mechanism to provide it an oscillatory movement; a parison transference mechanism mounted on said oscillatory hollow arm, including sliding carrier holders (327, 327') mounted on said oscillatory hollow arm, including holding arms normally maintained closed, and gripping fingers coupled to the holding arms for internally gripping the transferable and open-able neck ring mold; and a two-positions opening mechanism coupled to the carrier holders (327, 327'), so that, at a first position of the two-positions opening mechanism, the gripping fingers are partially opened in order to grip the transferable and open-able neck ring mold gripping a parison, while simultaneously an open-able horizontal neck ring holding arms of the indexed unidirectional rotary inverting apparatus are opened releasing said transferable and open-able neck ring mold, and be retained by the gripping fingers of this transference apparatus at the intermediate station, and at a second position, the holding arms are additionally opened, in order to additionally open the gripping fingers, opening the transferable and open-able neck ring mold releasing the parison to vertically fall flat into a blow mold apparatus, but keeping the transferable and open-able neck ring mold held by the gripping fingers; and in the inverse way, said gripping fingers of the transference apparatus return the empty transferable and open-able neck ring mold closed at the intermediate station, and are further closed, releasing the empty transferable and open-able neck ring mold at the open-able horizontal neck ring holding arms of the indexed unidirectional rotary inverting apparatus to be held thereby for a following forming cycle.