  Recent advancements in the synthesis of anisotropic macromolecules and
nanoparticles have spurred an immense interest in theoretical and computational
studies of self-assembly. The cornerstone of such studies is the role of shape
in self-assembly and in inducing complex order. The problem of identifying
different types of order that can emerge in such systems can, however, be
challenging. Here, we revisit the problem of quantifying orientational order in
systems of building blocks with non-trivial rotational symmetries. We first
propose a systematic way of constructing orientational coordinates for such
symmetric building blocks. We call the arising tensorial coordinates strong
orientational coordinates (SOCs) as they fully and exclusively specify the
orientation of a symmetric object. We then use SOCs to describe and quantify
local and global orientational order, and spatiotemporal orientational
correlations in systems of symmetric building blocks. The SOCs and the
orientational order parameters developed in this work are not only useful in
performing and analyzing computer simulations of symmetric molecules or
particles, but can also be utilized for the efficient storage of rotational
information in long trajectories of evolving many-body systems.
