  Random, uncorrelated displacements of particles on a lattice preserve the
hyperuniformity of the original lattice, that is, normalized density
fluctuations vanish in the limit of infinite wavelengths. In addition to a
diffuse contribution, the scattering intensity from the the resulting point
pattern typically inherits the Bragg peaks (long-range order) of the original
lattice. Here we demonstrate how these Bragg peaks can be hidden in the
effective diffraction pattern of independent and identically distributed
perturbations. All Bragg peaks vanish if and only if the sum of all probability
densities of the positions of the shifted lattice points is a constant at all
positions. The underlying long-range order is then 'cloaked' in the sense that
it cannot be reconstructed from the pair correlation function alone. On the one
hand, density fluctuations increase monotonically with the strength of
perturbations $a$, as measured by the hyperuniformity order metric
$\overline{\Lambda}$. On the other hand, the disappearance and reemergence of
long-range order, depending on whether the system is cloaked or not as the
perturbation strength increases, is manifestly captured by the $\tau$ order
metric. Therefore, while the perturbation strength $a$ may seem to be a natural
choice for an order metric of perturbed lattices, the $\tau$ order metric is a
superior choice. It is noteworthy that cloaked perturbed lattices allow one to
easily simulate very large samples (with at least $10^6$ particles) of
disordered hyperuniform point patterns without Bragg peaks.
