In this work we consider two families of piecewise-linear maps with a discontinuity, which is motivated by modeling of several power electronic circuits (DC/DC converters, Sigma-Delta modulators) and which is already investigated by many authors. However, the focus of these works lie on the investigation of the dynamics by variation of one or at most two system parameters. In many of these publications a great variety of bifurcation phenomena is revealed but unfortunately, a profound explanation of the complicated and often self-similar bifurcation structures and the systematics behind is missing. In contrast to that, we investigate the full 3D parameter space of these families of systems and detect a number of discontinuity induced codimension-3 bifurcations. Some of them belong to an already known generic type [1,2], while others represent some new types not investigated so far. Unfolding these bifurcations we are able to explain the systematics for a large number of the above mentioned bifurcation phenomena. In particular, several bifurcation scenarios observed under variation of one or two parameters represent an intersection of the extended bifurcation structures induced by codimension-3 bifurcation with the corresponding 1D or 2D parameter subspace. The codimension-3 bifurcations reported here are characterized by two manifolds in the 3D parameter space, a 1D and a 2D one. The 1D manifold represents a codimension-2 bifurcation curve, whereas in the 2D manifold an infinite number of codimension-2 bifurcation curves are located. At the codimension-3 bifurcation point all these curves intersect. The reported bifurcations serve as organizing centers of periodic and aperiodic dynamics in the overall parameter space. [1] V. Avrutin and M. Schanz, Nonlinearity 19, 531 (2006). [2] V. Avrutin, M. Schanz, and S. Banerjee, Nonlinearity 19, 1875 (2006).