Modulus of continuity of orientation preserving approximately differentiable homeomorphisms with a.e. negative Jacobian

We construct an a.e. approximately differentiable homeomorphism of a unit $n$-dimensional cube onto itself which is orientation preserving, has the Lusin property (N) and has the Jacobian determinant negative a.e. Moreover, the homeomorphism together with its inverse satisfy a rather general sub-Lipschitz condition, in particular it can be bi-H\"older continuous with an arbitrary exponent less than $1$.