  We study the rotational structures of aperiodic tilings in Euclidean space of
arbitrary dimension using topological methods. Classical topological approaches
to the study of aperiodic patterns have largely concentrated just on
translational structures, studying an associated space, the continuous hull,
here denoted $\Omega_t$. In this article we consider two further spaces
$\Omega_r$ and $\Omega_G$ (the rotational hulls) which capture the full rigid
motion properties of the underlying patterns. The rotational hull $\Omega_r$ is
shown to be a matchbox manifold which contains $\Omega_t$ as a sub-matchbox
manifold. We develop new S-MLD invariants derived from the homotopical and
cohomological properties of these spaces demonstrating their computational as
well as theoretical utility. We compute these invariants for a variety of
examples, including a class of 3-dimensional aperiodic patterns, as well as for
the space of periodic tessellations of $\mathbb{R}^3$ by unit cubes. We show
that the classical space group of symmetries of a periodic pattern may be
recovered as the fundamental group of our space $\Omega_G$. Similarly, for
those patterns associated to quasicrystals, the crystallographers' aperiodic
space group may be recovered as a quotient of our fundamental invariant.
