  A helical CR structure is a decomposition of a real Euclidean space into an
even-dimensional horizontal subspace and its orthogonal vertical complement,
together with an almost complex structure on the horizontal space and a marked
vector in the vertical space. We prove an equivalence between such structures
and step two Carnot groups equipped with a distinguished normal geodesic, and
also between such structures and smooth real curves whose derivatives have
constant Euclidean norm. As a consequence, we relate step two Carnot groups
equipped with sub-Riemannian geodesics with this family of curves. The
restriction to the unit circle of certain planar homogeneous polynomial
mappings gives an instructive class of examples. We describe these examples in
detail.
