  We study Lipschitz differentiability spaces, a class of metric measure spaces
introduced by Cheeger. We show that if an Ahlfors regular Lipschitz
differentiability space has charts of maximal dimension, then, at almost every
point, all its tangents are uniformly rectifiable. In particular, at almost
every point, such a space admits a tangent that is isometric to a
finite-dimensional Banach space. In contrast, we also show that if an Ahlfors
regular Lipschitz differentiability space has charts of non-maximal dimension,
then these charts are strongly unrectifiable in the sense of
Ambrosio-Kirchheim.
