We construct a Landau-Ginzburg model with the same data and symmetries as a
$Z_2\times Z_2$ orbifold that corresponds to a class of realistic free-fermion
models. Within the class of interest, we show that this orbifolding connects
between different $Z_2\times Z_2$ orbifold models and commutes with the mirror
symmetry. Our work suggests that duality symmetries previously discussed in the
context of specific $M$ and $F$ theory compactifications may be extended to the
special $Z_2\times Z_2$ orbifold that characterizes realistic free-fermion
models.
