  We introduce an axiom on strong parapolar spaces of diameter 2, which arises
naturally in the framework of Hjelmslev geometries. This way, we characterize
the Hjelmslev-Moufang plane and its relatives (line Grassmannians, certain
half-spin geometries and Segre geometries). At the same time we provide a more
general framework for a Lemma of Cohen, which is widely used to study parapolar
spaces. As an application, if the geometries are embedded in projective space,
we provide a common characterization of (projections of) Segre varieties, line
Grassmann varieties, half-spin varieties of low rank, and the exceptional
variety $\mathcal{E}_{6,1}$ by means of a local condition on tangent spaces.
