  P. Berglund, T. H\"ubsch, and M. Henningson proposed a method to construct
mirror symmetric Calabi-Yau manifolds. They considered a pair consisting of an
invertible polynomial and of a finite (abelian) group of its diagonal
symmetries together with a dual pair. A. Takahashi suggested a method to
generalize this construction to symmetry groups generated by some diagonal
symmetries and some permutations of variables. In a previous paper, we
explained that this construction should work only under a special condition on
the permutation group called parity condition (PC). Here we prove that, if the
permutation group is cyclic and satisfies PC, then the reduced orbifold Euler
characteristics of the Milnor fibres of dual pairs coincide up to sign.
