  We present three novel types of discrete Lagrangian 2-form for the integrable
quad equations of the ABS list. Two of our new Lagrangian 2-forms have the quad
equations, or a system equivalent to the quad equations, as their
Euler-Lagrange equations, whereas the third produces the tetrahedron equations.
This is in contrast to the well-established Lagrangian 2-form for these
equations, which produces equations that are weaker than the quad equations
(they are equivalent to two octahedron equations). We use relations between the
Lagrangian 2-forms to prove that the system of quad equations is equivalent to
the combined system of tetrahedron and octahedron equations. Furthermore, for
each of the Lagrangian 2-forms, existing and new, we study the double zero
property of the exterior derivative. In particular, this gives a possible
variational interpretation to the octahedron equations.
