  A framework, which is a (possibly infinite) graph with a realization of its
vertices in the plane, is called flexible if it can be continuously deformed
while preserving the edge lengths. We focus on flexibility of frameworks in
which 4-cycles form parallelograms. For the class of frameworks considered in
this paper (allowing triangles), we prove that the following are equivalent:
flexibility, infinitesimal flexibility, the existence of at least two classes
of an equivalence relation based on 3- and 4-cycles and being a non-trivial
subgraph of the Cartesian product of graphs. We study the algorithmic aspects
and the rotationally symmetric version of the problem. The results are
illustrated on frameworks obtained from tessellations by regular polygons.
