We prove that every quasisymmetric self-homeomorphism of the standard
1/3-Sierpi\'nski carpet $S_3$ is a Euclidean isometry. For carpets in a more
general family, the standard $1/p$-Sierpi\'nski carpets $S_p$, $p\ge 3$ odd, we
show that the groups of quasisymmetric self-maps are finite dihedral. We also
establish that $S_p$ and $S_q$ are quasisymmetrically equivalent only if $p=q$.
The main tool in the proof for these facts is a new invariant---a certain
discrete modulus of a path family---that is preserved under quasisymmetric maps
of carpets.