  A novel feature of the ${\mathbb Z}_2\times {\mathbb Z}_2$-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 ${\mathbb Z}_2\times
{\mathbb Z}_2$-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 ${\mathbb Z}_2\times {\mathbb Z}_2$-graded worldline
super-Poincar\'e algebra induces two-dimensional ${\mathbb Z}_2\times {\mathbb
Z}_2$-graded relativistic models; they are invariant under a new ${\mathbb
Z}_2\times {\mathbb Z}_2$-graded $2D$ super-Poincar\'e algebra which differs
from the previous two ${\mathbb Z}_2\times {\mathbb Z}_2$-graded $2D$ versions
of super-Poincar\'e 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 ${\bf S}^1$,
a ${\mathbb Z}_2\times {\mathbb Z}_2$-graded closed string with periodic
boundary conditions is derived. The analysis of the irreducibility conditions
of the $2D$ supermultiplet implies that a larger $(\beta$-deformed, where
$\beta\geq 0$ is a real parameter) class of point-particle models than the ones
discussed so far in the literature (recovered at $\beta=0$) is obtained. While
the spectrum of the $\beta=0$ point-particle models is degenerate (due to its
relation with an ${\cal N}=2$ supersymmetry), this is no longer the case for
the $\beta> 0$ models.
