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A symmetry of a dynamical system is a map that transforms one trajectory to another trajectory. We introduce a new type of abstraction for hybrid automata based on symmetries. The abstraction combines different modes in a concrete automaton A, whose trajectories are related by symmetries, into a single mode in the abstract automaton B. The abstraction sets the guard and reset of an abstract edge to be the union of the symmetry-transformed guards and resets of the concrete edges. We establish the soundness of the abstraction using a forward simulation relation (FSR) and present several examples. Our abstraction results in simpler automata, that are more amenable for formal analysis and design. We illustrate an application of this abstraction in making reachability analysis faster and enabling unbounded time safety verification. We show how a fixed point of the reachable set computation of B can be used to answer reachability queries for A, even if the latter visits an infinite and unbounded sequences of modes. We present our implementation of the abstraction construction, the fixed point check, and the map that transforms abstract reachable sets to concrete ones in a software tool. Finally, we show the advantage of our method over existing ones, and the different aspects of our abstraction, in a sequence of experiments including scenarios with linear and nonlinear agents following waypoints.