The vertices of the secondary polytope of a point configuration correspond
to its regular triangulations. The Cayley trick links triangulations of one
point configuration, called the Cayley polytope, to the fine mixed subdivi
sions of a tuple of point configurations. In this paper we investigate the
secondary polytope of this Cayley polytope. Its vertices correspond to all
regular mixed subdivisions of a tuple of point configurations. We demonstra
te that it equals the Minkowski sum of polytopes, which we call mixed secon
dary polytopes, whose vertices correspond to regular-cell configurations.