Many physical and engineerings systems exhibit cascades of periodic attractors arranged in period increment and period adding sequences as a parameter is varied. Such systems have been found to yield piecewise smooth maps, and in some cases the obtained map is discontinuous. By investigating the normal form of such maps, we have detected a new type of codimension-3 bifurcation which serves as the organizing center of periodic and aperiodic dynamics in the parameter space. The results will help in understanding the occurrence and structure of such cascades observed in many nonsmooth systems in science and engineering.