An orientation preserving homeomorphism of a cube may have a.e. negative Jacobian

In a joint work with Piotr Hajłasz, we construct an almost everywhere approximately differentiable (a.e.a.d) homeomorphism of a unit n-dimensional cube that is orientation and measure preserving, and at the same time its Jacobian determinant is equal to -1 almost everywhere.
By a similar construction, we obtain an a.e.a.d. homeomorphism that is orientation preserving, bi-Holder continuous with an arbitrary exponent a<1, and has negative Jacobian a.e.
These examples give answer to questions posed by Hajłasz in 2001.