Regularity of mappings inverse to Sobolev mappings

For homeomorphisms phi : Omega -> Omega' on Euclidean domains in R-n, n >= 2, necessary and sufficient conditions ensuring that the inverse mapping belongs to a Sobolev class are investigated. The result obtained is used to describe a new two-index scale of homeomorphisms in some Sobolev class such that their inverses also form a two-index scale of mappings, in another Sobolev class. This scale involves quasiconformal mappings and also homeomorphisms in the Sobolev class W-n-1(1) such that rank D phi(x) <= n-2 almost everywhere on the zero set of the Jacobian det D phi(x).