you're thinking of all boundaries mapping to a point. in your interpretation, the red circle moving to the boundary would look like this https://cdn.pixabay.co

you're thinking of all boundaries mapping to a point. in your interpretation, the red circle moving to the boundary would look like this https://cdn.pixabay.com/photo/2013/03/08/05/53/frame-91452_960_720.jpg https://cdn.pixabay.com/photo/2013/03/08/05/53/frame-91452_9... Instead, look at how the arrow types match here: https://upload.wikimedia.org/wikipedia/commons/3/37/Projection_color_torus.png https://upload.wikimedia.org/wikipedia/commons/3/37/Projecti... the side you choose to glue first matters (compare before/after) https://upload.wikimedia.org/wikipedia/commons/b/ba/Inside-out_torus_%28animated%2C_small%29.gif https://upload.wikimedia.org/wikipedia/commons/b/ba/Inside-o... and the orientation of the arrows https://upload.wikimedia.org/wikipedia/commons/6/64/Klein_Bottle_Folding_1.svg https://upload.wikimedia.org/wikipedia/commons/6/64/Klein_Bo... also matters https://upload.wikimedia.org/wikipedia/commons/5/5c/Klein_bottle.svg https://upload.wikimedia.org/wikipedia/commons/5/5c/Klein_bo...