  In a series of papers, Burdzy et. al. introduced the \emph{mirror coupling}
of reflecting Brownian motions in a smooth bounded domain $D\subset
\mathbb{R}^{d}$, and used it to prove certain properties of eigenvalues and
eigenfunctions of the Neumann Laplaceian on $D$. In the present paper we show
that the construction of the mirror coupling can be extended to the case when
the two Brownian motions live in different domains $D_{1},D_{2}\subset
\mathbb{R}^{d}$. As an application of the construction, we derive a unifying
proof of the two main results concerning the validity of Chavel's conjecture on
the domain monotonicity of the Neumann heat kernel, due to I. Chavel
(\cite{Chavel}), respectively W. S. Kendall (\cite{Kendall}).
