I have a question regarding when $M^*$ is the oriented doublel covering of $M$. Suppose for example $M$ is the Mobius strip. As it is non-orientable, $M^*$ should be connected. Indeed from some pictures that I saw, it is path connected. Look at the picture in this post Orientable double-cover of the Mobius strip by John Hughes.
(update: be wary of the wrong definition in the next paragraph!)
Now as a set, I denote $M^*$ by $\{(p, +_p)| p \in M\} \cup \{(p, -_p)| p \in M\}$ where here $+_p$ denotes the equivalence class of all positive orientations on $T_pM$ and $-_p$ is defined similarly. My confusion is as follows. If $\gamma:[0, 1]\rightarrow M^*$ is a curve connecting $(p, +_p)$ to $(q, -_q)$, at which point does a positive orientation (the $+$) change to a negative orientation (to a $-$)? This is very weired to me. Even when I look at the drawing of the double cover, there seems to be something weired about it. It is indeed a double cover and it is orientable. But I still can't correspond this mathematical construction to the picture I see!
update: The following answer https://math.stackexchange.com/a/2893383/148256 deals with a correct definition.