Play with the white points.
Here you can play hyperbolizations of a ornament of the type P2 (in orbifold notation 2222)
with rectangular fundamental cell.
Here the ornament does not completely determine the symmetry type.
After fixing the corner angles the symmetry still has a free parameter that can be adjusted by the white point within the hyperbolic picture.
The corner angles of the hyperbolic fundamental cell can be adjusted with the sliders.