Symmetries and integrability of discrete equations defined on a black-white lattice

We study the deformations of the H equations, presented recently by Adler, Bobenko and Suris, which are naturally defined on a black-white lattice. For each one of these equations, two different three-leg forms are constructed, leading to two different discrete Toda-type equations. Their multidimensional consistency leads to Backlund transformations relating different members of this class as well as to Lax pairs. Their symmetry analysis is presented yielding infinite hierarchies of generalized symmetries.