I have seen that a double covering of the Klein bottle is the torus and basically a polygonal representation of the covering is the following. We consider the standard polygonal representation of the torus and we identify the two edges of the torus with a line in the middle of the rectangle oriented opposite to the orientations of the edges. What gets me confused is that I don't understand why this shape is still a torus. I mean for example if we were to identify the two edges with a line in the middle having the same orientation as the edges then the shape wouldn't be a torus it would be homeomorphic to two tori glued in the top. So why does the identification with the middle line in opposite direction doesn't disturb the torus.
Thanks in advance