  The concept of hyperuniformity has been a useful tool in the study of
large-scale density fluctuations in systems ranging across the natural and
mathematical sciences. One can rank a large class of hyperuniform systems by
their ability to suppress long-range density fluctuations through the use of a
hyperuniformity order metric $\bar{\Lambda}$. We apply this order metric to the
Barlow packings, which are the infinitely degenerate densest packings of
identical rigid spheres that are distinguished by their stacking geometries and
include the commonly known fcc lattice and hcp crystal. The "stealthy stacking"
theorem implies that these packings are all stealthy hyperuniform, a strong
type of hyperuniformity which involves the suppression of scattering up to a
wavevector $K$. We describe the geometry of three classes of Barlow packings,
two disordered classes and small-period packings. In addition, we compute a
lower bound on $K$ for all Barlow packings. We compute $\bar{\Lambda}$ for the
aforementioned three classes of Barlow packings and find that to a very good
approximation, it is linear in the fraction of fcc-like clusters, taking values
between those of least-ordered hcp and most-ordered fcc. This implies that the
$\bar{\Lambda}$ of all Barlow packings is primarily controlled by the local
cluster geometry. These results indicate the special nature of anisotropic
stacking disorder, which provides impetus for future research on the
development of anisotropic order metrics and hyperuniformity properties.
