No, your cross-analogue would mostly work the OTHER way
CO2 ice on Mars would behave more like H2O ice on Earth (with the notable exception of a liquid phase). CO2 ice on Mars falls as snow every winter, Thaws (sublimes) in the spring. CO2 ice forms as frost in cold areas, and just fades away when the sun comes out.
CO2 is the "active" volatile on Mars. H2O is closer to a refractory mineral...
