A "double" Möbius Strip

This is more or less the same question as Why aren't the "higher twist" Möbius bands distinct line-bundles over $S^1$?

A "Möbius-like strip with two (half) twists" is topologically a cylinder.

In fact you have to distinguish between the cylinder $C = S^1 \times [0,1]$ as a topological space which is obtained as the quotient space described in your question and the embeddings of $C$ into $\mathbb R^3$. You can embed $C$ with an arbitrary number $n$ of (full) twists into $\mathbb R^3$ and for $n \ne n'$ these embeddings cannot be deformed into each other via an isotopy.