- Geometrical symmetries are commonly exploited to improve the efficiency of search algorithms. A new type of symmetry in permutation state spaces, duality, is introduced. Each state has a dual state. Both states share important attributes such as their distance to the goal. Given a state S, it is shown that an admissible heuristic of the dual state of S is an admissible heuristic for S. This provides opportunities for additional heuristic evaluations. An exact definition of the class of problems where duality exists is provided. A new search algorithm, dual search, is presented which switches between the original state and the dual state when it seems likely that the switch will improve the chance of reaching the goal faster. The decision of when to switch is very important and several policies for doing this are investigated. Experimental results show significant improvements for a number of applications, for using the dual state's heuristic evaluation and/or dual search.