Blaschke Products Sharing a Welding Homeomorphism

Given two Blaschke products $A$, $B$ of the same degree, a \emph{welding homeomorphism} of $A$, $B$ is a circle homeomorphism $k:\mathbb{T}\to \mathbb{T}$ satisfying $A\circ k=B$ on $\mathbb{T}$. We show that any two pairs of Blaschke products sharing a welding homeomorphism must factor through a common pair of Blaschke products, and we give a criterion for detecting whether two Blaschke products admit such a factorization.