  We show that given a homeomorphism $f:G\rightarrow\Omega$ where $G$ is a open
subset of $\mathbb{R}^2$ and $\Omega$ is a open subset of a $2$-Ahlfors regular
metric measure space supporting a weak $(1,1)$-Poincar\'e inequality, it holds
$f\in BV_{\operatorname{loc}}(G,\Omega)$ if and only $f^{-1}\in
BV_{\operatorname{loc}}(\Omega,G)$. Further if $f$ satisfies the Luzin N and
N$^{-1}$ conditions then $f\in W^{1,1}_{\operatorname{loc}}(G,\Omega)$ if and
only if $f^{-1}\in W^{1,1}_{\operatorname{loc}}(\Omega,G)$.
