The object of study of this paper is the class of hybrid systems consisting of so-called linear complementarity (LC) systems, that received a lot of attention recently and has strong connections to piecewise affine (PWA) systems. In addition to PWA systems, some of the linear or affine submodels of the LC systems can 'live' at lower-dimensional subspaces and re-initializations of the state variable at mode changes is possible. For LC systems we study the stability and controllability problem. Although these problems received for various classes of hybrid systems ample attention, necessary and sufficient conditions, which are explicit and easily verifiable, are hardly found in the literature. For LC systems with two modes and a state dimension of two such conditions are presented.