= Solution
If $(x,y)\sim(x+a,y+b)$ in the torus, then
$$
\widetilde\pi(x+a,y+b)=(2x+2a,y+b).
$$
This is Klein-equivalent to $(2x,y)$ by taking $(c,d)=(2a,b)$. Hence $\widetilde\pi$ descends to a well-defined continuous map
$$
\pi:T^2\to K,
\qquad
\pi([x,y])=[2x,y].
$$
The deck <group> of the Klein bottle has an index-two <subgroup> consisting of transformations with even $c$; it is generated by translations $(x,y)\mapsto(x+2,y)$ and $(x,y)\mapsto(x,y+1)$. This is exactly the image under $\widetilde\pi$ of the torus deck lattice. Therefore the <orientation double cover of the Klein bottle> has two sheets:
$$
\boxed{\pi:T^2\to K\text{ is }2:1}.
$$
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