I read somewhere recently that when you cut the Klein bottle in two you get two Möbius strips that twist in opposite directions. But I think that's not true. The picture shows that both of the line segments twist the same way, whichever way the propeller is turning as you carry them around.

And this makes sense, because when you make the Möbius strip, the endpoints of the line segment make a helix as you carry them around, And if you want to glue two helices together, they have to twist in the same direction, not different directions.

I think when you cut the bottle you get a choice about which way the two Möbius strips twist, but they both have to be the same.

(Note that twist direction is a property of the immersion, not intrinsic.)