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This is NOT the same as the fundamental group of the Klein bottle as well --- the fundamental group of Klein bottle has normal subgroup $\mathbb Z^2$ inside and no $\mathbb Z^2$ in $\pi_1$ of your orbifold.

If you want to think in terms of loops, then choose a local lift of each loop in every chart of some atlas on such a way that they agree by some of transition maps. Then think about homotopy with such liftings.

In case your orbifold is spin and oriented and dim $\ge 3$, you may also pass to the double cover of frame bundle and think there.

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This is NOT the same as the fundamental group of the Klein bottle as well --- the fundamental group of Klein bottle has normal subgroup $\mathbb Z^2$ inside and no $\mathbb Z^2$ in $\pi_1$ of your orbifold.

If you want to think in terms of loops, then choose a local lift of each loop in every chart of some atlas on such a way that they agree by some of transition maps. Then think about homotopy with such liftings.

In case your orbifold is spin and oriented and dim $\ge 3$, you may also pass to the double cover of frame bundle and think there.