Both passive and active mechanisms are necessary to explain small amplitude forward-backward (FB) voluntary swaying. Parallel and symmetric leg inverted pendulum models with stiffness control are a simple way to replicate FB swaying during quiet stance. However, it has been more difficult to model lateral left-right (LR) voluntary swaying involving the dual mechanisms of hip loading-unloading and ankle pressure distribution. To assess these factors, we had subjects perform small amplitude FB and LR sways and circular rotation. We experimentally identified three parameters that characterized their two-dimensional stiffnesses: AP stiffness (K), and lateral stiffness (K), at the ankles and a parameter we refer to as the engagement-disengagement rate (K) of the legs. We performed simulations with our engaged leg model (Bakshi A, DiZio P, Lackner JR. 121: 2042-2060, 2019; Bakshi A, DiZio P, Lackner JR. 121: 2028-2041, 2019) to test its predictions about the limits of balance stability during sway in the three test conditions. Comparing the model's predictions with the experimental data, we found that K has a task-dependent dual role in upright balance and is crucial to prevent falling; K helps overcome viscous drags but is not instrumental to stability; K has a key role in stability and is dependent on the biomechanical geometry of the body, which is invariant across balance tasks. These findings provide new insights into balance control that have important clinical implications for falling, especially for patients who are unable to use a hip strategy during balance control. Our previously published Engaged Leg Model here shows how stiffness plays complex multicausal roles in balance. In one role, it is crucial to stability, with task contingent influences over balance. In another, it overcomes viscous drag. Task-dependent stiffness alone does not explain stable balance; geometrical, invariant aspects of body biomechanics also matter. Our model is fully applicable to clinical balance pathologies involving asymmetries in movement and balance control.