Let $\mathcal{C}$ denote the space of great circles in $\mathbb{S}^2\subset \mathbb{R}^3$. It's pretty easy to see that any element $\mathcal{C}$ can be identified uniquely with the axial line (in $\mathbb{R}^3$) to the circle and so $\mathcal{C}$ can be identified with $\mathbb{RP}^2$. If $\mathcal{OC}$ denotes the space of oriented great circles, then one also can check that $\mathcal{OC}$ can be identified with $\mathbb{S}^2$ and the natural double cover $\pi': \mathcal{OC}\to \mathcal{C}$ given by forgetting orientation is compatible with the usual double cover $\pi: \mathbb{S}^2\to \mathbb{RP}^2$. Explicitly, one has for each $\mathbf{v}\in \mathbb{S}^2$ the oriented great circle $C(\mathbf{v})\in \mathcal{OC}$ given by $\{\mathbf{y}\in \mathbb{S}^2: \mathbf{y}\cdot \mathbf{v}=0\}$ and oriented by counter-clockwise rotation about $\mathbf{v}$.
Now let $\mathcal{PC}$ denote the space of pointed great circles in $\mathbb{S}^2$, i.e., the set of pairs $(\mathbf{x}, C)$ so $x\in C$ and $C\in \mathcal{C}$. Likewise let $\mathcal{POC}$ be the space of pointed oriented great circles. $\mathcal{POC}$ can be identified with $SO(3)$ and hence with $\mathbb{RP}^3$. For instance, consider a $(\mathbf{x}, C)\in \mathcal{POC}$. Write $C=C(\mathbf{v})$ and observe that, as $\mathbf{x}\in C$, $\mathbf{x}\cdot \mathbf{v}=0$. The map given by $(x,C(\mathbf{v}))\mapsto [\mathbf{x} | \mathbf{v} | \mathbf{x}\times \mathbf{v}]$ can readily be checked to be the desired bijection with $SO(3)$.
My question is: In analogy with the first situation, is there a space $X$ with a nice geometric description as a configuration space of circles with extra structure and a forgetful double cover $\pi': X\to \mathcal{POC}$ so that $X$ can be identified with $Spin(3)$ and this is compatible with the double cover $\pi: Spin(3)\to SO(3)$ (or $\pi: \mathbb{S}^3\to \mathbb{RP}^3$)?
