  We analyze subsets of Carnot groups that have intrinsic constant normal, as
they appear in the blowup study of sets that have finite sub-Riemannian
perimeter. The purpose of this paper is threefold. First, we prove some mild
regularity and structural results in arbitrary Carnot groups. Namely, we show
that for every constant-normal set in a Carnot group its
sub-Riemannian-Lebesgue representative is regularly open, contractible, and its
topological boundary coincides with the reduced boundary and with the
measure-theoretic boundary. We infer these properties from a cone property.
Such a cone will be a semisubgroup with nonempty interior that is canonically
associated with the normal direction. We characterize the constant-normal sets
exactly as those that are arbitrary unions of translations of such
semisubgroups. Second, making use of such a characterization, we provide some
pathological examples in the specific case of the free-Carnot group of step 3
and rank 2. Namely, we construct a constant normal set that, with respect to
any Riemannian metric, is not of locally finite perimeter; we also construct an
example with non-unique intrinsic blowup at some point, showing that it has
different upper and lower sub-Riemannian density at the origin. Third, we show
that in Carnot groups of step 4 or less, every constant-normal set is
intrinsically rectifiable, in the sense of Franchi, Serapioni, and Serra
Cassano.
