We obtain normal forms for symmetric and reversible polynomial automorphisms (polynomial maps that have polynomial inverses) of the complex and real planes. Our normal forms are based on the Hénon normal form of Friedland and Milnor. We restrict ourselves to the case where the symmetries and reversors are also polynomial automorphisms. We show that each such reversor has finite order and that for nontrivial, real maps, the reversor has order 2 or 4. The normal forms are shown to be unique up to finitely many choices. We investigate some of the dynamical consequences of reversibility, especially for the case where the reversor is not an involution.