Summary: Let \(f_ n = \sum^ n_{k=1}v_ kr_ k\), \(n = 1,\dots,\) be a martingale transform of a Rademacher sequence \((r_ n)\) and let \((r_ n')\) be an independent copy of \((r_ n)\). The main result of this paper states that there exists an absolute constant \(K\) such that for all \(p\), \(1\leq p < \infty\), the following inequality is true: \[ \bigl\| \sum v_ kr_ k\bigr\|_ p\leq K\bigl\|\sum v_ kr_ k'\bigr\|_ p. \] In order to prove this result, we obtain some inequalities which may be of independent interest. In particular, we show that for every sequence of scalars \((a_ n)\) one has \[ \bigl\| \sum a_ kr_ k\bigr\|_ p \approx K_{1,2}((a_ n),\sqrt{p}), \] where \(K_{1,2}((a_ n),\sqrt{p}) = K((a_ n),\sqrt{p};\ell_ 1,\ell_ 2)\) is the \(K\)-interpolation norm between \(\ell_ 1\) and \(\ell_ 2\). We also derive a new exponential inequality for martingale transforms of a Rademacher sequence.