  In this paper, we examine the magnetoelectric response of Ising pyrochlores,
focusing on both the ordered antiferromagnetic state and the frustrated
ferromagnetic case known as "spin-ice". We employ a model which accounts for
magnetoelastic effects by considering the interplay between oxygen distortions
and superexchange magnetic interactions within pyrochlores. This, together with
numerical simulations, provides a tool to make quantitative comparisons with
experiments. Our main target is then to see how to extract relevant information
from this simple model, and to explore its limitations. We obtain a direct
estimation of quantities such as the electric dipole moment, the central oxygen
displacement and the effective magnetoelastic energy for the canonical spin-ice
material \DTO. We also inquire about the possibility of using the electric
dipole carried by magnetic monopoles to obtain a direct measure of their
density. In each studied scenario the correlations between monopoles, induced
by their number or by the magnetic background,renders these findings less
straightforward than initially anticipated. Furthermore, the coupling between
electrical and magnetic degrees of freedom provides additional tools to
investigate magnetic order in these systems. As an example of this we discuss
the phase diagram of the antiferromagnetic pyrochlore under applied magnetic
field along the [111] direction. We find an instance where the phase stability
at nonzero temperatures is not dictated by the energy associated with different
ground states but (akin to the phenomenon of order-by-disorder) is instead
determined by their accessibility to thermal fluctuations.
