The embedded singly periodic Scherk surfaces of genus 0 are all classified, but it is nevertheless instructive what is happening if one changes the parameters.
Above are embedded examples that have two horizontal annular ends, while the remaining ends make an angle of 30º and 89º degrees with the horizontal plane. They also have the vertical coordinate planes as symmetry planes. When the angles approaches 0, we get a simple noded limit. Increasing the angle to 90º and higher creates non-embedded surfaces with another symmetry type.
They have a straight line in the direction of the translational axis. It would be interesting to find a desingularization of the two intersections.