A new treatment is given of the cylinder-web diagram and associated diagonal sequences in homotopy pair theory. The efficiency of the diagram as a machine for computing homotopy pair groups is enhanced by a result that traces the path of a Toda bracket element through the arrows of the diagram. The diagonal factorization problem for a homotopy pair class is studied and related to the behaviour of Toda brackets. A necessary and sufficient condition for the vanishing of a Toda bracket is obtained.