Let $X\subset \mathbb R^2$ be compact and connected. My question is whether homeomorphisms of $X$ preserve boundaries of complementary components.

More precisely, let $h:X\to X$ be a homeomorphism. If $U$ is a component of $\mathbb R^2\setminus X$ and $B=\partial U$, does there exist a component $V$ of $\mathbb R^2\setminus X$ such that $h[B]=\partial V$?

I am mostly interested in the case when $X$ is locally connected and $1$-dimensional. Under these assumptions, $B$ is a locally connected continuum, and is essentially a simple closed curve.