In this paper, we develop a new necessary and sufficient condition for the vanishing of -Massey products of elements in the modulo- Galois cohomology of a field. This new description allows us to define a splitting variety for -Massey products, which is shown in the appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such -Massey products vanish whenever they are defined. This provides new explicit restrictions on th...