  We investigate the alternate order on a congruence-uniform lattice
$\mathcal{L}$ as introduced by N. Reading, which we dub the core label order of
$\mathcal{L}$. When $\mathcal{L}$ can be realized as a poset of regions of a
simplicial hyperplane arrangement, the core label order is always a lattice.
For general $\mathcal{L}$, however, this fails. We provide an equivalent
characterization for the core label order to be a lattice. As a consequence we
show that the property of the core label order being a lattice is inherited to
lattice quotients. We use the core label order to characterize the
congruence-uniform lattices that are Boolean lattices, and we investigate the
connection between congruence-uniform lattices whose core label orders are
lattices and congruence-uniform lattices of biclosed sets.
