Let F denote a free group. An n-Engel word of F is an iterated commutator of the form [A, nB], where (A,B) is a pair of elements of F, which we here assume to generate a non-cyclic subgroup of F. This is the first of a series of three papers that yield a description of some of the properties — asphericity, word problem, local nilpotence — of a group defined by a suitable set of relators of the form [Ai, nBi], i = 1, 2, ..., m (where n is fixed). The present paper develops and extends ideas introduced by the second author in [3], involving the use of cancellation diagrams, to pro- vide the technical machinery for application in the subsequent companion papers.