There are several different patterns available for Klein Bottle hats. I used one (not currently available) by Debbie New, with a few modifications, to knit a hat to take to a Maths conference. It is different from most of the others because it is in garter stitch with the rows going vertically. One of the people at the conference was metagrobologist David Singmaster. He fell in love with the hat so I gave it to him ... then made another exactly the same to replace it.
The orange and blue hat was made, in the same way, for a young Maths performer from Circo Matemático.
A Klein Bottle can be made from two Möbius strips so I adapted the pattern to create two strangely-shaped strips. To be sure of the construction I first made the hat in paper, then in fabric, then cut it apart and joined it back together again. The knitting is easy. The most difficult part is finding the way to turn the completed knitting to create the hat shape.