I would like to characterize the set of all continuous $G:[0,1]^n \rightarrow [0,1]^n$ such that the Jacobian determinant is one:
$\mathcal{G}=\{G:[0,1]^n \rightarrow [0,1]^n: |DG(x)|=1\forall x \}$.
Is it true that $\mathcal{G}$ corresponds to the set of all linear transformations $G(x)=Ax+b$ with orthogonal matrices $A$? Indeed we have $DG= A$ in this case. And taking a determinant equal to one should imply that $A$ is orthogonal.
On
As implied by @Chrystomath's comment, these maps are interesting to those studying incompressible material deformation. For example, this work$^1$ describes a family of such maps with generating functions.
Considering a bounded region $\Omega_0$ in the n-dimensional Euclidean space, this work is aimed at defining generating functions for volume-preserving transformations of a set of continuously differentiable functions $u_j = u_j(U_1, U_2, \ldots, U_n): \Omega_0 \to \mathfrak{R}$, with $j = 1,2,\ldots,n$. Such an isochoric constraint can be expressed by a non-linear first-order partial differential equation as follows:
$$ J(U_1, U_2, \ldots, U_n) = \text{det} \frac{\partial(u_1, u_2, \ldots, u_n)}{\partial(U_1, U_2, \ldots, U_n)} = 1 $$ where $J$ is defined as the Jacobian of the transformation.
This is a relaxation of your definition, since your $\mathcal{G}$ set is defined using a codomain of $[0, 1]^n$, the same as your domain. The cited work considers maps $\Omega_0 \to \mathfrak{R}$, not $\Omega_0 \to \Omega_0$. If you are strictly interested in the latter, then this is only of limited relevance. If not, then the generating functions in the work are relevant. For instance, equation (13) and examples in equation (14) and Figure 3.
First, you need your functions to be $C^1$ (or at least have 1st order partial derivatives exist) for the Jacobian to be defined.
Second, $(-1)$ is an orthogonal $1 \times 1$ matrix, but $G(x) = -x$ does not define a function $[0,1] \to [0,1]$, so you conjecture is false for that reason.
Why do you believe that there is a nice classification of such maps?
Third, I would note that you really only want to be talking about differentiability on open sets, though as long as you want continuous differentiability, I think you'll be fine.
Fourth, I can't think of any nontrivial (as in not the identity map) functions with Jacobian 1 that map $I^n \to I^n$. Can you give an example? There might not be any.