This page shows the details for the construction of the first 6 simplist Hexaflexagons. These maps are all based on the 1962 paper by A. S. Conrad which can be found here:
See, Hexaflexagon Construction , for an explanation of these diagrams. The first set of maps are for the three simplest hexaflexagons. For each of the following 3, 4, and 5 sideded hexaflexagons, there is only one variation.
Now for the 6 sided Hexaflexagons. In this case there are three variations and all of them are quite interesting! Here they are: