  Insulating spinel materials, with the chemical formula $AB_2X_4$, behave as
diamond lattice antiferromagnets when only the A-site atom is magnetic. Many
exhibit classic signatures of frustration, induced not geometrically but by
competing first and second neighbor exchange interactions. In this paper, we
further develop a theory of the magnetism of these materials, focusing on the
physics observable within the ordered state. We derive a phenomenological
Landau theory that predicts the orientation of the spins within incommensurate
spiral ordered states. It also describes how the spins reorient in a magnetic
field, and how they may undergo a low temperature "lock-in" transition to a
commensurate state. We discuss microscopic mechanisms for these magnetic
anisotropy effects. The reduction of the ordered moment by quantum fluctuations
is shown to be enhanced due to frustration. Our results are compared to
experiments on MnSc_2S_4, the best characterized of such A-site spinels, and
more general implications are discussed. One prediction is that
magnetically-induced ferroelectricity is generic in these materials, and a
detailed description of the relation of the electric polarization to the
magnetism is given.
