This paper introduces a method for achieving stable periodic walking for legged robots. This method is based on producing a type of odd-even symmetry in the system. A hybrid system with such symmetries is called a Symmetric Hybrid System (SHS). We discuss the properties of an SHS and, in particular, will show that an SHS can have an infinite number of synchronized periodic orbits. We describe how controllers can be obtained to make a legged robot an SHS. Then the stability of the synchronized periodic orbits of this SHS is studied, where the notion of self-synchronization is introduced. We show that such self-synchronized periodic orbits are neutrally stable in kinetic energy. As the final step in the process of achieving asymptotically stable periodic walking, we show how by introducing asymmetries (such as energy loss at impact) in the system , the synchronized periodic orbits of this SHS can be turned into asymptotically stable periodic orbits. Many numerical examples, including an 8-DOF 3D biped with 2 degrees of underactuation, are studied to demonstrate the effectiveness of the method.