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In previous investigations into the subject [Giuntini et al. (2007, Studia Logica, 87, 99–128), Paoli et al. (2008, Reports on Mathematical Logic, 44, 53–85), Bou et al. (2008, Soft Computing, 12, 341–352)], ' quasi-MV algebras have been mainly viewed as preordered structures w.r.t. the induced preorder relation of their quasi-MV term reducts. In this article, we shall focus on a different relation which partially orders cartesian ' quasi-MV algebras. We shall prove that: (i) every cartesian ' quasi-MV algebra is embeddable into an interval in a particular Abelian l-group with operators; (ii) the category of cartesian ' quasi-MV algebras isomorphic with the pair algebras over their own polynomial MV subreducts is equivalent both to the category of such l-groups (with strong order unit), and to the category of MV algebras. As a by-product of these results we obtain a purely group-theoretical equivalence, namely between the mentioned category of l-groups with operators and the category of Abelian l-groups (both with strong order unit).