We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpi'nski carpet into the plane. A complete characterization in the case of so called dyadic slit carpets is given. Every such slit carpet $X$ can be embedded into a ``pillowcase sphere" $\widehat{X}$ which is a metric space homeomorphic to the sphere $\mathbb{S}^2$. We show that $X$ can be quasisymmetrically embedded into the plane if and only if $\widehat{X}$ is quasisymmetric to $\mathbb{S}^2$ if and only if $\widehat{X}$ is Ahlfors $2$-regular. The main tools used are Schramm’s transboundary modulus and the quasisymmetric uniformization theorem of Bonk-Kleiner.