  The Erd\H{o}s--Gallai criteria for recognizing degree sequences of simple
graphs involve a system of inequalities. Given a fixed degree sequence, we
consider the list of differences of the two sides of these inequalities. These
differences have appeared in varying contexts, including characterizations of
the split and threshold graphs, and we survey their uses here. Then, enlarging
upon properties of these graph families, we show that both the last term and
the maximum term of the principal Erd\H{o}s--Gallai differences of a degree
sequence are preserved under graph complementation and are monotonic under the
majorization order and Rao's order on degree sequences.
