A novel feature of the Z 2 ˆZ2 -graded supersymmetry which finds no counterpart in ordinary supersymmetry is the presence of 11-graded exotic bosons (implied by the existence of two classes of parafermions). Their interpretation, both physical and mathematical, presents a challenge. The role of the "exotic bosonic coordinate" was not considered by previous works on the one-dimensional Z 2 ˆZ2 -graded superspace (which was restricted to produce point-particle models). By treating this coordinate at par with the other graded superspace coordinates new consequences are obtained. The graded superspace calculus of the Z 2 ˆZ2 -graded worldline super-Poincaré algebra induces two-dimensional Z 2 ˆZ2 -graded relativistic models; they are invariant under a new Z 2 ˆZ2 -graded 2D super-Poincaré algebra which differs from the previous two Z 2 ˆZ2graded 2D versions of super-Poincaré introduced in the literature. In this new superalgebra the second translation generator and the Lorentz boost are 11-graded. Furthermore, if the exotic coordinate is compactified on a circle S 1 , a Z 2 ˆZ2 -graded closed string with periodic boundary conditions is derived. The analysis of the irreducibility conditions of the 2D supermultiplet implies that a larger pβ-deformed, where β ě 0 is a real parameter) class of point-particle models than the ones discussed so far in the literature (recovered at β " 0) is obtained. While the spectrum of the β " 0 point-particle models is degenerate (due to its relation with an N " 2 supersymmetry), this is no longer the case for the β ą 0 models.

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