Abstract. We create a framework for hereditary graph classes [Formula: see text] built on a two-dimensional grid of vertices and edge sets defined by a triple [Formula: see text] of objects that define edges between consecutive columns, edges between nonconsecutive columns (called bonds), and edges within columns. This framework captures a large family of minimal hereditary classes of graphs of unbounded clique-width, some previously identified and many new ones, although we do not claim this includes all such classes. We show that a graph class [Formula: see text] has unbounded clique-width if and only if a certain parameter [Formula: see text] is unbounded. We further show that [Formula: see text] is minimal of unbounded clique-width (and, indeed, minimal of unbounded linear clique-width) if another parameter [Formula: see text] is bounded, and also [Formula: see text] has defined recurrence characteristics. Both the parameters [Formula: see text] and [Formula: see text] are properties of a triple [Formula: see text] and measure the number of distinct neighborhoods in certain auxiliary graphs. Throughout our work, we introduce new methods to the study of clique-width, including the use of Ramsey theory in arguments related to unboundedness, and explicit (linear) clique-width expressions for subclasses of minimal classes of unbounded clique-width.