This surface appears to be the simplest among Peter Connor’s experimental doubly periodic genus 3 surfaces.
But appearance can deceive. First, there is a complicated orientation reversing symmetry: Reflect at a horizontal plane, then slide horizontally and diagonally by halfperiods. Next one wonders how the catenoidal handles will move. The above image is already close to one limit, where the neck sizes shrink to zero while the handles are being pushed towards each other, before ever reaching a symmetric position. The other limit is more straightforward, showing two doubly periodic Karcher-Scherk surfaces.