  We consider lattice equations on ${\mathds{Z}}^2$ which are autonomous,
affine linear and possess the symmetries of the square. Some basic properties
of equations of this type are derived, as well as a sufficient linearization
condition and a conservation law. A systematic analysis of the Lie point and
the generalized three- and five-point symmetries is presented. It leads to the
generic form of the symmetry generators of all the equations in this class,
which satisfy a certain non-degeneracy condition. Finally, symmetry reductions
of certain lattice equations to discrete analogues of the Painlev\'e equations
are considered.
