  Ebeling and Ploog \cite{EbelingPloog} studied a duality of bimodular
singularities which is part of the Berglund--H$\ddot{\textnormal{u}}$bsch
mirror symmetry. Mase and Ueda \cite{MU} showed that this duality leads to a
polytope mirror symmetry of families of $K3$ surfaces. We discuss in this
article how this symmetry extends to a symmetry between lattices.
