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A novel feature of the ${\mathbb Z}_2\times {\mathbb Z}_2$-gradedsupersymmetry which finds no counterpart in ordinary supersymmetry is thepresence of $11$-graded exotic bosons (implied by the existence of two classesof parafermions). Their interpretation, both physical and mathematical,presents a challenge. The role of the "exotic bosonic coordinate" was notconsidered by previous works on the one-dimensional ${\mathbb Z}_2\times{\mathbb Z}_2$-graded superspace (which was restricted to producepoint-particle models). By treating this coordinate at par with the othergraded superspace coordinates new consequences are obtained. The gradedsuperspace calculus of the ${\mathbb Z}_2\times {\mathbb Z}_2$-graded worldlinesuper-Poincar\'e algebra induces two-dimensional ${\mathbb Z}_2\times {\mathbbZ}_2$-graded relativistic models; they are invariant under a new ${\mathbbZ}_2\times {\mathbb Z}_2$-graded $2D$ super-Poincar\'e algebra which differsfrom the previous two ${\mathbb Z}_2\times {\mathbb Z}_2$-graded $2D$ versionsof super-Poincar\'e introduced in the literature. In this new superalgebra thesecond translation generator and the Lorentz boost are $11$-graded.Furthermore, if the exotic coordinate is compactified on a circle ${\bf S}^1$,a ${\mathbb Z}_2\times {\mathbb Z}_2$-graded closed string with periodicboundary conditions is derived. The analysis of the irreducibility conditionsof the $2D$ supermultiplet implies that a larger $(\beta$-deformed, where$\beta\geq 0$ is a real parameter) class of point-particle models than the onesdiscussed so far in the literature (recovered at $\beta=0$) is obtained. Whilethe spectrum of the $\beta=0$ point-particle models is degenerate (due to itsrelation with an ${\cal N}=2$ supersymmetry), this is no longer the case forthe $\beta> 0$ models.

New aspects of the Z$_{\textrm 2}$ $\times$ Z$_{\textrm 2}$-graded 1D superspace: induced strings and 2D relativistic models