Square symmetric composite laminate structures, and sub-modules thereof, are provided, along with methods of forming the same. The square symmetric laminate structures include two or more sub-laminate modules, each comprising: a first ply set consisting of a first ply layer oriented at a first angle and a second ply layer oriented at a second angle, a first sum of the first and second angles being ninety degrees; and a second ply set consisting of a third ply layer oriented at a third angle and a fourth ply layer oriented at a fourth angle, a second sum of the third and fourth angles being ninety degrees; wherein the second ply layer is positioned adjacent the third ply layer and the second and third ply layers are both positioned intermediate the first and fourth ply layers, thereby defining a double-double helix arrangement of the respective ply layers. Associated methods are also provided.