The image $M=\gamma(S)$ is a Mobius Strip, what are the missing points of the parameterization of the interior and what is another parameterization?

Question

Let $$S=\{(r,\theta) | -\frac12 <r<\frac12 , 0\leq\theta<2\pi \}$$ and $$S'=\{(r,\theta) | -\frac12 <r<\frac12 , 0<\theta<2\pi \}$$ so $S'$ is the interior of $S$. Define $\gamma : S \rightarrow \mathbb{R}^3$ by: $$\gamma(r,\theta) = \begin{bmatrix}2cos\theta+rcos(\frac\theta2)\\2sin\theta+rsin(\frac\theta2)\\rsin(\frac\theta2)\end{bmatrix}$$ Then the image $M=\gamma(S)$ is a Mobius Strip. We may assume that $\gamma$ is injective on $S$ and $D(\gamma)(\overrightarrow c)$ is injective for all $\overrightarrow c \in S$ without having to prove this. But then this makes $\gamma$ an invalid parametrization because $S$ is not open, so we restrict the domain to $S'$. Which points of $M$ are missing from the image $\gamma(S')$? Also, how would I find another open set $T\subseteq\mathbb{R}^2$ and another parameterization $\phi:T\rightarrow M$ such that $\phi(T)$ is the "most of" M (in the same sense that $\gamma(S')$ is most of M), but the set of points missing from $\phi(T)$ is different from the set of points missing from $\gamma(S')$ and describe the missing points of M. (We don't need to prove that $\phi$ and $D(\phi)$ are injective)