Fibers and textiles are ubiquitous in our daily lives, with mechanical properties that match the design specifications for the task for which they are intended; the development of yarns with a negative Poisson’s ratio (NPR) is a hot topic of current research, owing to their potential for use in high-performance textiles (e.g., military, sports, etc.). This study described a simple approach to constructing braided, helically interlaced yarns. When a torque is applied, the yarns prevent the wrapped component from dislodging from the core. The geometry and auxetic behavior of the braided helical structure was analyzed for two different combinations of core materials with similar wrap materials and different braiding angles. Two elastomeric materials (polyurethane (PU) and polyester) served as monofilament cores, while two stiffer multifilament wrap yarns (ultrahigh molecular weight polyethylene (UHMWPE) and polyethylene terephthalate (PET)) served as wrap yarns. In addition, the behavior of yarns braided at seven different angles was investigated to determine the materials’ response to the applied braided configuration’s NPR. The NPR was influenced by the core and wrap materials used and the braiding angle. The NPR value was greater for a core comprising more excellent elasticity (e.g., PU versus polyester); a smaller wrap angle and a slower braiding speed also led to a higher NPR value. The maximum NPR value of −1.70 was obtained using a PU core wrapped at a 9° angle and a strain value of 0.5.