Klein Bottle:- tricky to come up with something which maintains a similar visual representation but how about following the contour with a tight travelling pirouetting loop whose radii closes down thus forming a horizontal helix. When the radii is minimal, more or less piroflipping, move into a tight piro half loop to form the right hand part of the Klein. Follow through into the folded-in portion still maintaining the klein profile. As you get to the left hand side, wind out into a piro tight piro loop and you back at where you started.
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