Ferroelectric ceramics are widely used as sensors and actuators for their electro-mechanical properties, and in electronic applications for their dielectric properties. Domain switching--the phenomenon wherein the ferroelectric material changes from one spontaneously polarized state to another under electrical or mechanical loads--is an important attribute of these materials. However, this is a complex collective process in commercially used polycrystalline ceramics that are agglomerations of a very large number of variously oriented grains. As the domains in one grain attempt to switch, they are constrained by the differently oriented neighbouring grains. Here we use a combined theoretical and experimental approach to establish a relation between crystallographic symmetry and the ability of a ferroelectric polycrystalline ceramic to switch. In particular, we show that equiaxed polycrystals of materials that are either tetragonal or rhombohedral cannot switch; yet polycrystals of materials where these two symmetries co-exist can in fact switch.