  We consider substitution tilings in R^d that give rise to point sets that are
not bounded displacement (BD) equivalent to a lattice and study the cardinality
of BD(X), the set of distinct BD class representatives in the corresponding
tiling space X. We prove a sufficient condition under which the tiling space
contains continuously many distinct BD classes and present such an example in
the plane. In particular, we show here for the first time that this cardinality
can be greater than one.
