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The top response add's rigor to of the OP's observation that the double Torus with a line down the middle gives rise to two klein bottles, but the top response also says that the diagram is proof enough. Can somebody help me gain some intuition as to why this is? Why can we just draw a line down the middle? Is that an elementary transformation and thus produces a topologically equivalent structure?

If that is true, does it mean that the torus is topologically equivalent to two Klein bottles with the edge in the diagram they share identified together? Then for the covering map we should just map each of these Klein bottles in the cut torus to a Klein bottle? Am i missing something here?