Deterministic thermostats are frequently employed in nonequilibrium molecular dynamics simulations in order to remove the heat produced irreversibly over the course of such simulations. The simplest thermostat is the Gaussian thermostat, which satisfies Gauss's principle of least constraint and fixes the peculiar kinetic energy. There are of course infinitely many ways to thermostat systems, e.g., by fixing [summation]i|pi|嫱. In the present paper we provide, for the first time, convincing arguments as to why the conventional Gaussian isokinetic thermostat (影) is unique in this class. We show that this thermostat minimizes the phase space compression and is the only thermostat for which the conjugate pairing rule holds. Moreover, it is shown that for finite sized systems in the absence of an applied dissipative field, all other thermostats (囮ot-equal]1) perform work on the system in the same manner as a dissipative field while simultaneously removing the dissipative heat so generated. All other thermostats (囮ot-equal]1) are thus autodissipative. Among all 場hermostats, only the 影 Gaussian thermostat permits an equilibrium state.