An essentially passive and "fuel-less" method for inverting the orientation of a preferably nutationally stable, dual spin spacecraft (10) disposed in an inclined orbit, includes the steps of increasing the rotational speed of (i.e., "spinning up") the spacecraft's despun platform (20) and decreasing the rotational speed of (i.e., "spinning down") the spacecraft's rotor (16), to thereby generate, via product of inertia coupling, a transverse torque of sufficient magnitude to temporarily destabilize the spacecraft and cause the spacecraft spin axis (12), which is the minimum moment of inertia axis of the spacecraft, to diverge and precess through a flat spin orientation and towards a final, inverted orientation, e.g., disposed at a precession angle of 180° relative to the initial orientation of the spacecraft spin axis. Normally, prior to the spacecraft spin axis reaching the final, inverted orientation, the spin axis encounters and is "stuck at" a barrier nutation angle beyond which further precession of the spacecraft spin axis is not possible without the implementation of a final acquisition procedure, which includes the steps of detecting the occurrence of the spin axis reaching the barrier nutation angle, and in response thereto, instituting limit cycle rotational motion of the platform, to thereby render non-secular the rotational motion of the platform, whereby the spacecraft damping system is brought into its pull-in or effective operating range for penetrating the barrier nutation angle, in order to achieve acquisition of the final inverted orientation of the spacecraft spin axis. Limit cycle rotation of the platform is preferably effectuated by means of applying a first and a second series of torquing pulses to the spacecraft despin motor suitable for reducing the rotational speed of the platform, the first series of pulses being of larger magnitude than that of the second series of pulses.