From each Lie group $G$ and principal $G$-bundle $P \rightarrow E$ one can form an associated (or gauge) Lie groupoid as the quotient of pair groupoid by the action of $G$ on $P \times P$ $$ \frac{P \times P}{G} \rightrightarrows B$$
In particular in case of the double cover of $SO(3)$ $$\mathbb Z/2 \rightarrow SU(2)\rightarrow SO(3) $$
Identifying $SU(2)$ with unit quaternions and $\mathbb R^4$ with quaternions $\mathbb H$, for each $(x, y) \in SU(2) \times SU(2)$ the map $v \mapsto xvy^{-1}$ defines a well known double cover of Lie groups $$SU(2) \times SU(2) \rightarrow SO(4)$$ Thus the associated groupoid of this bundle has a form $$SO(4) \rightrightarrows SO(3)$$
Are there any reasonable geometric interpretations of multiplication in this groupoid?