Watching this video, I found a resemblance between the Klein bottle and the inside out bill as described by Bob Neale in The Sense of Wonder, p187, under the title of The Tortured Bill (also in some videos by Ralf Rudolph).

https://www.youtube.com/watch?v=u_xQ_C5RUfo

Is my observation correct? I understand all of the edges would have to be glued together.