Belyi functions that permute vertex colors

The canonical triangulation of a dessin has vertices of three colors: black for original vertices, white for edge centers, 'star' for face centers. The simplest Belyi functions that permute these colors in the 6 possible ways are the 6 transformations Coxeter, in Regular Polytopes, gave as an example of the operation of the symmetric group on three elements:

Here's what they look like in geographic domain coloring (geographic conventions same as previous post.) All these functions describe a half-edge or brin in different positions/orientations: its black vertex is in each case coincident with Antarctica, its white vertex coincident with Null Island.

Identity, z:

Dual, 1/z:

1-z:

z/(z-1):

1/(1-z):

(z-1)/z: