Much of the recent analysis of chaotic mixing has focused on utilizing tools and techniques imported from dynamical systems theory. However, most techniques require detailed information about the velocity field or fluid motion and are restricted to conditions where the `degree of chaos' is small. Symmetries provide a method of analysis without specific reference to exact mathematical expressions. Symmetry concepts are illustrated in terms of a prototypical system called the eggbeater flow. Although a family of 32 different eggbeater flows can be constructed, symmetry arguments reveal that only four of these are independent. These flows serve to illustrate the role of islands in mixing. If a flow possesses symmetry, islands are found in symmetric arrangements, the simplest cases being reflectional and rotational symmetries. A knowledge of symmetries provides the basis for systematic methods for destroying islands. These ideas are developed in terms of the eggbeater flows, and are subsequently extended to a class of three-dimensional continuous throughput flows - duct flows - which are of a more practical interest from an engineering viewpoint. Three such duct flows are studied. Using symmetries, we show that these flows are topologically identical to the eggbeater flows, even though their geometries and flow mechanisms are quite different from the eggbeater flows. Lastly, we demonstrate how the same methodology for destroying islands and enhancing mixing in the eggbeater flows can be applied to duct flows.