Palaeomagnetic studies of Palaeoproterozoic to Cretaceous rocks propose a suite of large and relatively rapid (tens of degrees over 10 to 100 million years) excursions of the rotation pole relative to the surface geography, or true polar wander (TPW). These excursions may be linked in an oscillatory, approximately coaxial succession about the centre of the contemporaneous supercontinent. Within the framework of a standard rotational theory, in which a delayed viscous adjustment of the rotational bulge acts to stabilize the rotation axis, geodynamic models for oscillatory TPW generally appeal to consecutive, opposite loading phases of comparable magnitude. Here we extend a nonlinear rotational stability theory to incorporate the stabilizing effect of TPW-induced elastic stresses in the lithosphere. We demonstrate that convectively driven inertia perturbations acting on a nearly prolate, non-hydrostatic Earth with an effective elastic lithospheric thickness of about 10 kilometres yield oscillatory TPW paths consistent with palaeomagnetic inferences. This estimate of elastic thickness can be reduced, even to zero, if the rotation axis is stabilized by long-term excess ellipticity in the plane of the TPW. We speculate that these sources of stabilization, acting on TPW driven by a time-varying mantle flow field, provide a mechanism for linking the distinct, oscillatory TPW events of the past few billion years.