By Richard Schoen’s theorem, it is not possible to make a genus one catenoid, but one can make a genus one surface with two Enneper ends. In the most symmetric case, one still has a 1-parameter family.
Versions with higher dihedral symmetry exist, too. I wonder what happens to the handle when one rotates the two Enneper ends as in the twisted version of the Double Enneper surface. In particular, can one achieve a full 360º twist?
Resources
PoVRay Sources
Minimal Surfaces 