Yes, there is such a thing as a hexaflexagon. I'm not sure what it's practical applications might be, but it's pretty fascinating to watch it come together!

 

The first hexaflexagon was made by a British graduate student, Arthur H. Stone, in 1939, as he was playing with some strips of paper in class.

If you would like to make your own hexaflexagon, instructions are here and here.