$(INV)$ condition and regularity of the inverse

Let $f\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ be a Sobolev mapping of finite distortion between planar domains $\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\mathrm{\prime }}$, satisfying the $(INV)$ condition and coinciding with a homeomorphism near $\mathrm{\partial }\mathrm{\Omega }$. We show that $f$ admits a generalized inverse mapping $h\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}^{\mathrm{\prime }}\to \mathrm{\Omega }$, which is also a Sobolev mapping of finite distortion and satisfies the $(INV)$ condition. We also establish a higher-dimensional analogue of this result: if a mapping $f\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ of finite distortion is in the Sobolev class $W^{1,p}(\mathrm{\Omega },{\mathbb{R}}^n)$ with $p>n-1$ and satisfies the $(INV)$ condition, then $f$ has an inverse in $W^{1,1}({\mathrm{\Omega }}^{\mathrm{\prime }},{\mathbb{R}}^n)$ that is also of finite distortion. Furthermore, we characterize Sobolev mappings satisfying $(INV)$ whose generalized inverses have finite $n$-harmonic energy.