This paper presents an analysis of coupled longitudinal and lateral dynamics of a 4×4 hybrid-electric off-road vehicle (HEV) with two passive driveline systems, including drivelines with (i) an interaxle open symmetrical differential in the transfer case and (ii) a locked transfer case, i.e., positive engagement of two axles. The axle differentials are open. As the study proved, lateral dynamics of the 4×4 HEV, characterized by the tire side forces, vehicle lateral acceleration, yaw rate and tire gripping factors can be impacted by the tire longitudinal forces, whose magnitudes and directions (positive-negative) strongly depend on the driveline characteristics. At the same time, the tire side forces impact the relation between the longitudinal forces and tire slippages. Thus, coupled dynamics of the vehicle in its longitudinal and lateral directions results in a strong interdependence between the vehicle’s mobility, estimated in the longitudinal direction, and the vehicle’s stability in the lateral direction. The paper presents mathematical models of the 4×4 HEV vehicle and its driveline systems. A detailed mathematical apparatus describes an analysis of a new vehicle mobility index as a function of the longitudinal force distribution between the wheels and the tires’ gripping factors that characterize the lateral skid of the tires and vehicle. It is shown that in certain conditions, the vehicle can be immobilized due to increased tire slippages caused by the tire side forces. Vice versa, the vehicle can go into a lateral skid due to increased tire side forces that are caused by the longitudinal force distribution between the wheels even when the vehicle velocity is small. Computational results illustrate the coupled dynamics and provide recommendations for decoupling the impact of the driveline systems on vehicle mobility and stability of motion.