The bottle was first theoretically described in 1882 by Felix Klein, hence the name.
I found that on the same website where several of my above pics originated.
They do sell Klein bottles there, if anyone's interested, but I thought their pitch was pretty funny, especially the part about mouse traps.
"At last, Acme has conquered topological and engineering frontiers to manufacture genuine glass Klein Bottles. These are the finest closed, non-orientable, boundary-free manifolds sold anywhere in our three spatial dimensions.
These elegant bottles make splendid gifts, outstanding classroom displays, and inferior mouse-traps
. With its circle of singularities, an Acme Klein Bottle can be said to exist inside of itself -- especially handy during time-reversals.
A mathematician's delight, handcrafted in glass"