In this paper a systematic study of bisimilarities on asymmetric \chi ^ \ne -processes is carried out. The notion of L- bisimilarities on asymmetric \chi ^ \ne -processes is introduced. Twelve distinct L-bisimilarities are derived from all of L- bisimilarities by constructing a bisimulation lattice. For each of these twelve distinct L-bisimilarities, its open version is defined and showed to coincide with it, and then its congruence is presented. Three update laws are proposed and three tau laws are modified. Finally, sound complete equational systems are established for twelve congruences.