This was a toughy. If you've never seen a Klein Bottle before, it's hard to describe in words or even in pictures. Imagine if you will, an object with only one side and no corners and no inside and no outside. There's no possible way to lay this item flat because it's connected to itself everywhere.
Here's a glass depiction:
(Incidentally, the artist sells these glass creations on Etsy)
And here's the scarf I came up with that's the best I could come up with in a day. I made all the surfaces real so even though you can't see it, there's a disc of fabric trapped where the thinner loop and the thicker part of the bottle meet. Finishing was a bear.