This genus 5 surface can be interpreted as a twisted version of the oS deformation of the Schwarz P surface.
The divisor of the squared Gauss map on the quotient torus is very similar but corresponds to a different spin structure.
The parameters a and b satisfy a+a’=1/2=b+b’ and a+b=1/4. The latter condition is responsible for the quarter twist. There is a single period condition left that determines τ in terms of a. One can probably relax the symmetries and get a larger deformation family.
The two limits of this family are a node plane configuration and doubly periodic Scherk surfaces stitched together using catenoidal necks. This is one of the simplest examples I know where the latter type of limit occurs, it has not been analyzed.