Describing coordinate surfaces

Question

Some arbitrary coordinate system with components $x'^a$ is related to Cartesian coordinates, $x^a$, by \begin{align} x^1 &= x'^1 +x'^2 \\\ x^2 &= x'^1 - x'^2 \\\ x^3 &= 2x'^1x'^2 + x'^3. \end{align} What are the coordinate surfaces described by the primed system? I understand that the coordinate surfaces of the Cartesian system are flat planes where one of the coordinates is fixed (e.g. $x=1$). I have rearranged the equations above to give the primed coordinates in terms of Cartesian: \begin{align} x'^1 &= \frac{1}{2}(x^1 + x^2) \\ x'^2 &= \frac{1}{2}(x^1 - x^2) \\ x'^3 &= x^3 - \frac{1}{2}\left((x^1)^2 - (x^2)^2\right). \end{align} I am struggling to visualise what fixing one of the primed coordinates, say $x'^1$, and letting the others vary would look like. My educated guess would say that they also describe 2D planes, but not mutually orthogonal.