Geometric constraints associated with the concave oceanward bend in the trench and Andes Mountains, known as the Bolivian Orocline, force along-strike compression in the subducting slab. Our calculations of membrane strain rates, in an assumed continuous slab, demonstrate that this can be accommodated by either 10% along-strike compressive strain, or by geometric buckling. In nearly all other subduction zones exhibiting a similar bend in the trench, the compression is accommodated by slab-arching landward of the trench bend. The slab geometry is markedly different in South America. Two regions of nearly flat-lying slab north and south of the Bolivian Orocline result in the 150-km slab depth contour being much straighter than the trench profile. Our calculations show that this geometry helps to relieve along-strike compression. Along the deeper part of the subduction zone, the along-strike compression seems to be accommodated by pronounced along-strike buckling. We infer the approximate geometry of this buckle, in an otherwise aseismic region, from the great Mw 8.3 Bolivian deep earthquake on June 9, 1994, and its aftershocks. The deep slab appears to have a nearly east-west strike in the vicinity of this earthquake, whereas the strike is more nearly north-south to the north and south. The focal mechanisms of deep earthquakes exhibit consistent down-dip compression and along-strike null axes, tracking the contortions in slab geometry. However, our calculations indicate that the particle paths themselves are not down dip everywhere. Instead, where the deep-slab strike is oriented nearly east-west, the particle flow is more nearly along strike than down dip. In general this is not a major problem, but if the slab is anchored in the lower mantle, and moving very slowly, this produces large deformation rates near the base of the upper mantle where the slab is assumed to move at near plate rates. This increased, complex deformation in the deep slab may contribute to the conditions for very large earthquakes in two ways. First, this may dramatically increase deformation rates, and secondly, the increased deformation may lead to anomalous advective thickening of a proposed overdriven olivine wedge. This provides a mechanism for locally thickening the olivine wedge and allowing transformational faulting to occur over a much larger volume than allowed for normal subduction.