  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 B{\"a}cklund transformations relating different members of
this class, as well as to Lax pairs. Their symmetry analysis is presented
yielding infinite hierarchies of generalized symmetries.
