  The cross-ratios do not uniquely fix the class of conformally equivalent
configurations of null polygons. In view of applications to Wilson loops and
scattering amplitudes we characterise all conformal classes of null hexagon
configurations belonging to given points in cross-ratio space. At first this is
done for the ordered set of vertices. Including the edges, we then investigate
the equivalence classes under conformal transformations for null hexagons. This
is done both for the set of null hexagons closed in finite domains of Minkowski
space as well as for the set including those closed via infinity.
