We compute the elliptic genus for arbitrary two dimensional $N=2$
Landau-Ginzburg orbifolds. This is used to search for possible mirror pairs of
such models. We show that if two Landau-Ginzburg models are conjugate to each
other in a certain sense, then to every orbifold of the first theory
corresponds an orbifold of the second theory with the same elliptic genus (up
to a sign) and with the roles of the chiral and anti-chiral rings interchanged.
These orbifolds thus constitute a possible mirror pair. Furthermore, new pairs
of conjugate models may be obtained by taking the product of old ones. We also
give a sufficient (and possibly necessary) condition for two models to be
conjugate, and show that it is satisfied by the mirror pairs proposed by one of
the authors and~H\"ubsch.