= Solution
Let $D=[0,1]\times[-1,1]$ and $q=p\circ i:D\to S/\mathcal R'$. This is <continuous>. It is onto: given $(x,y)\in S$, choose an <integer> shift bringing $x$ into $[0,1]$, with the corresponding possible sign change in $y$; the resulting representative lies in $D$.
Two representatives in $D$ are in the same <orbit> only if their first coordinates differ by an <integer>. An interior point can therefore be equivalent only to itself. Distinct <boundary> representatives have first coordinates zero and one, and the iterate $g$ identifies exactly $(0,y)$ with $(1,-y)$. Thus the fibres of $q$ are precisely the <equivalence classes> defining the <Möbius band> $M$.
The <rectangle> $D$ is <compact>, and the target is <Hausdorff> by part (i). Applying part (a)(v) to $q$ therefore yields
$$
\boxed{M=D/\mathcal R\cong S/\mathcal R'.}
$$
The <compactness> argument supplies <continuity> of the inverse without requiring a discontinuous choice of one representative per <orbit>.
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