There’s a component for this in Kangaroo now:
The hexagons are always flat because Möbius transformations (which inversions are) preserve circles. Since the initial regular hexagons have all their vertices lying on a circle, so do the transformed ones, therefore they are planar.
Also - here’s another covering of the sphere with only 6 sided patches -
fold a hexagon in half along a line joining a pair of opposite vertices, glue the sides of the boundary to each other, and inflate it:
