does this glued object exist?

Question

Say there is a square G with four vertices $(a,b,c,d)$. From the diagonal direction of the square, I try to glue the edges so that the directions of the arrows match. I am curious whether the object that follows the glue rules in the below figure: In the last step, I don't think the object is a Sphere and I could not image how the object will look like. I think it does not exist: the vertex $a$, $c$ and $b$ will be glued together while edges $ab$ and $bc$ are glued inversely, contradicting each other. But maybe I am wrong? If it exists, how will it look like? thank you very much!

Answer 1

This way of trying to glue the boundaries does not work (because the edges don't have matched orientations). But you can glue $(ab)$ to $(dc)$ and then $(da)$ to $(bc)$. This yields a Klein Bottle (why?).