How to build a Möbius Kaleidocycle

First fold the hinge lines as valley folds.



...and so on...



Then fold all the other lines as moutain folds. There are two groups of parallel lines.
Group 1:



...and so on...
Then group 2:



...and so on...



Then start gluing.



...and so on for 3 and 4...



Do not glue the 6 yet!



Glue 5 and close the chain.



Congratulations, you built a Möbius Kaleidocycle!