A Klein bottle is like a Mobius strip, only with one more dimension. Just as a Mobius strip has only one side, a Klein bottle has only one surface - there is no inside!
I made this small Polarfleece version as a gift for a mathematically inclined friend, and then realized it would make a good hat (modeled here by Boc, my living room tyranosaurus):
I actually made a human-size version, but it is in darker fleece and so wouldn't photograph as well.
Here is a tute. I do not have a very steady hand at Paint, so pretend that all the wiggly lines are smooth curves outlining shapes with top/bottom symmetry. The basic pattern piece looks like an hourglass that has been stretched out so that one lobe is taller than the other:
Maybe the width doesn't actually have to be a whole 36". If each lobe has 6" or so of wide space for the head to fit into, that seems to be enough.
Because of the topographical properties of the Klein bottle, it needs only two seams (plus the optional dart, which makes the hat a nicer shaped cone). Don't make the narrow neck too narrow to yank the rest of it through in the unfolding process.
Also, because there is no inside, you should use a fabric that looks good on both sides. The raw edge will show on the neck, so best to use fleece or some other fabric such that a raw edge doesn't look too bad. I guess you could make it a fringy edge...
To accentuate the no-inside-or-outside part, I ran a ribbon along the length of it and tied it to itself as a kind of road map.
Come winter, I may start cranking out Klein bottle polarfleece hats with matching Mobius strip scarves. Thank you, Polarfleece!
P.S. The prototype in the photo doesn't have a raw edge showing on the neck because I messed up and had to cut, twist, and re-sew its neck (I hid the extra seam within the bottle). But for the larger version I did it the way the tute describes, and it works, honest!