  The Chern-Moser normal form and its analog on finite type hypersurfaces in
general do not respect symmetries. Extending the work of N. K. Stanton, we
consider the local equivalence problem for symmetric Levi degenerate
hypersurfaces of finite type in $\mathbb C^2$. The results give for all such
hypersurfaces a complete normalization which respects the symmetries. In
particular, they apply to tubes and rigid hypersurfaces, providing an effective
classification. The main tool is a complete normal form constructed for a
general hypersurface with a tube model. As an application, we describe all
biholomorphic maps between tubes, answering a question posed by N. Hanges.
