  In this paper we continue the study of the notion of
$\mathscr{P}$-rectifiability in Carnot groups. We say that a Radon measure is
$\mathscr{P}_h$-rectifiable, for $h\in\mathbb N$, if it has positive $h$-lower
density and finite $h$-upper density almost everywhere, and, at almost every
point, it admits a unique tangent measure up to multiples. In this paper we
prove a Marstrand--Mattila rectifiability criterion in arbitrary Carnot groups
for $\mathscr{P}$-rectifiable measures with tangent planes that admit a normal
complementary subgroup. Namely, in this co-normal case, even if a priori the
tangent planes at a point might not be the same at different scales, a
posteriori the measure has a unique tangent almost everywhere.
  Since every horizontal subgroup of a Carnot group has a normal complement,
our criterion applies in the particular case in which the tangents are
one-dimensional horizontal subgroups. Hence, as an immediate consequence of our
Marstrand--Mattila rectifiability criterion and a result of
Chousionis--Magnani--Tyson, we obtain the one-dimensional Preiss's theorem in
the first Heisenberg group $\mathbb H^1$. More precisely, we show that a Radon
measure $\phi$ on $\mathbb H^1$ with positive and finite one-density with
respect to the Koranyi distance is absolutely continuous with respect to the
one-dimensional Hausdorff measure $\mathcal{H}^1$, and it is supported on a
one-rectifiable set in the sense of Federer, i.e., it is supported on the
countable union of the images of Lipschitz maps from $A\subseteq \mathbb R$ to
$\mathbb H^1$.
