Automotive suspensions invariably exhibit asymmetric damping properties in compression and rebound, which is partly attributed to asymmetric damping and in-part to the suspension linkage kinematics together with tire lateral compliance. Although automotive suspensions have invariably employed asymmetric damping, the design guidelines and particular rationale for such asymmetry has not been explicitly defined. The influences of damper asymmetry together with the suspension kinematics and tire lateral compliance on the dynamic responses of a vehicle are investigated analytically under bump and pothole excitations, and the results are interpreted in view of potential design guidance. A quarter-car kineto-dynamic model of the road vehicle employing a double wishbone type suspension comprising a strut with linear spring and multiphase asymmetric damper is formulated for the analyses. The simulation results revealed conflicting sprung mass acceleration responses under idealized bump and pothole road inputs. The results attained from a sensitivity analysis suggested significant influences of damper asymmetry, and the compression and rebound reduction factors corresponding to higher strut speeds on the dynamic responses. A composite performance index comprising the ride comfort, rattle space and tire road holding properties of the vehicle is formulated to seek optimal damping asymmetry. The optimal suspension damping parameters derived through minimization of the composite index function revealed considerable potential for improved ride responses under the bump and pothole excitations.