  We develop a theory of quasisymmetries for finitely ramified fractals, with
applications to finitely ramified Julia sets. We prove that certain finitely
ramified fractals admit a naturally defined class of "undistorted metrics" that
are all quasi-equivalent. As a result, piecewise-defined homeomorphisms of such
a fractal that locally preserve the cell structure are quasisymmetries. This
immediately gives a solution to the quasisymmetric uniformization problem for
topologically rigid fractals such as the Sierpi\'nski triangle. We show that
our theory applies to many finitely ramified Julia sets, and we prove that any
connected Julia set for a hyperbolic unicritical polynomial has infinitely many
quasisymmetries, generalizing a result of Lyubich and Merenkov. We also prove
that the quasisymmetry group of the Julia set for the rational function
$1-z^{-2}$ is infinite, and we show that the quasisymmetry groups for the Julia
sets of a broad class of polynomials contain Thompson's group $F$.
