  A sum-and-distance system is a collection of finite sets of integers such
that the sums and differences formed by taking one element from each set
generate a prescribed arithmetic progression. Such systems, with two component
sets, arise naturally in the study of matrices with symmetry properties and
consecutive integer entries. Sum systems are an analogous concept where only
sums of elements are considered. We establish a bijection between sum systems
and sum-and-distance systems of corresponding size, and show that sum systems
are equivalent to principal reversible cuboids, which are tensors with integer
entries and a symmetry of "reversible square" type. We prove a structure
theorem for principal reversible cuboids, which gives rise to an explicit
construction formula for all sum systems in terms of joint ordered
factorisations of their component set cardinalities.
