  We prove that every quasisymmetric homeomorphism of a standard square
Sierpi\'nski carpet $S_p$, $p\ge 3$ odd, is an isometry. This strengthens and
completes earlier work by the authors. We also show that a similar conclusion
holds for quasisymmetries of the double of $S_p$ across the outer peripheral
circle. Finally, as an application of the techniques developed in this paper,
we prove that no standard square carpet $S_p$ is quasisymmetrically equivalent
to the Julia set of a postcritically-finite rational map.
