Klein bottle homeomorphic to union of Möbius strip

Question

I'm having trouble showing that the Klein bottle defined as a quotient space of $I^2$ with relation $(x,-1)R(x,1)$ and $(-1,y)R(1,-y)$ is Hausdorff and that it can be expressed as $X\cup Y$ where $X,Y$ are homeomorphic to the Möbius strip and $X\cap Y$ is homeomorphic to $S^1$.