In topology, we learn that when you cut a klein bottle into two, you get two Möbius strips. In other words, a connected sum of two strips. Do these strips have the same or opposite chirality? Is there an easy way to visualize it?
No, the strips are mirror images.
Yes. See https://i.stack.imgur.com/2sPTw.png
Two Mobius strips of opposite chirality are one possible outcome of "cutting" a Klein bottle "into two". Depending on how the "cutting" is done, you can end up with two cylinders or two triangles with all three vertices attached together. (A more familiar example: Take a torus. Depending on how you "cut" it "into two", you can get a cylinder, two annuli, or two triangles with all three vertices attached together.)
On the number of Klein bottle types by Carlo H. Séquin (Journal of Mathematics and the Arts, Volume 7, Issue 2, 2013)
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