Let $S^3$ be a unit cube in $\mathbb{R}^3$ with the boundary points identified. Let $\chi$ be a continuous, surjective map from $S^3$ onto $SO(3)$ where the preimage $\chi^{-1}(x)$ has a cardinality of 2 for every $x \in SO(3)$. From my intuitive understanding this should correspond to a homeomorphism to the double cover of $SO(3)$, $S^3 \rightarrow Spin(3)$ and therefore exist even though a one folded map (homeomorphism) $S^3 \rightarrow SO(3)$ does not exist.
Is this intuitive understanding correct, i.e. does such a continuous map exist? If so, do you know of any visualization of that?