In this talk, we will discuss uniqueness and symmetry results for steady gradient Ricci solitons that are asymptotically quotient-cylindrical. Under a rigidity assumption, we show that the steady solitons of Bryant and Appleton are unique among solitons with the same asymptotics. In dimension 4, we show that under the same rigidity hypothesis, any asymptotically quotient-cylindrical steady soliton contains a circle symmetry. We prove these results by establishing a symmetry principle which also generalizes to expanding solitons and Ricci-flat ALE spaces.