Disordered materials, such as glasses or granular media, exist in a rugged potential-energy landscape with a myriad of stable configurations that defy easy enumeration and characterization. Introducing physical symmetries into these systems can decrease the number of available states, resulting in a smoother energy landscape. In this talk, I will show how a specific type of symmetry, namely permutation symmetry in disordered packings of soft spherical particles, leads to remarkable order in the distribution of basin volumes on the energy landscape. Breaking this symmetry leads to additional, distinct configurations for arranging the particles, thus rendering the landscape more rugged. I will present a novel algorithm that uncovers signs of the regularity initially observed in the landscape with intact symmetry, within the landscape with broken symmetry. The principles discussed in this talk are extendable to various geometrical symmetries in disordered systems.