We introduce a novel class of quadrilateral gridshell structures in axial force equilibrium where rods are aligned symmetrically with the principal stress directions of a limit membrane shell. These structures exhibit a distinctive property where the axial forces in the four connected rods at each node are nearly equal. This characteristic enables a more uniform distribution of forces within the structure, particularly in cases where stresses exhibit significant anisotropy. In contrast, conventional gridshells often result in numerous rods remaining nearly unloaded in such scenarios. We begin by studying the equilibrium of rod networks that are symmetric to principal stress directions. Next, we explore the geometric properties of these networks in relation to the isotropic geometry of their Airy stress surface. We introduce then a computational pipeline for designing principal symmetric structures through quadrilateral remeshing of a surface in membrane equilibrium and subsequent optimization. Finally, we present some of the achieved results.