Quasisymmetric extension on the real line

We give a geometric characterization of the sets $E\subset \mathbb{R}$ that satisfy the following property: every quasisymmetric embedding $f: E \to $\mathbb{R}$^n$extendstoaquasisymmetricembedding$f:$\mathbb{R}$\to$\mathbb{R}$^N$forsome$N\geq n$.