I have tried numerous times to come up with a surface that could proudly call itself the Diamond Surface with a Handle, in vain. It turns out this is easier with two handles.
The identifications are by horizontal translations and a vertical translation together with a diagonal translation by a half period.
The divisor of the squared Gauss map on the quotient torus is shown above. The parameters a and b need to satisfy a+b=1/2, and there is a 1-dimensional period problem to solve. The translation structures of G dh and 1/G dh are show below.
The two limits consist of noded planes at one end, and doubly periodic Karcher-Scherk surfaces at the other.
In between, the surfaces become surprisingly tall.