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. 

Hexaflexagon maps 3-5


 Now for the 6 sided Hexaflexagons.  In this case there are three variations and all of them are quite interesting!  Here they are:


Six sided maps