The symmetrizations of the Enneper surface appear in Hermann Karcher’s Tokyo notes, but were possibly known earlier.
The Weierstrass representation is
for n+1-fold symmetry. They are, like Enneper’s surface, intrinsically rotationally symmetric. I have one curious conjecture about them: If you parametrize them in polar coordinates as in the images, there is a certain radius up to which they are embedded. This radius appears to be the the 2n-th root of an algebraic number of degree n-1. For instance, for n=4, the radius is the 8th root of the largest solution of . See the notebook for more.
Mathematica notebook for conjecture