Materials exhibiting a coupled response to both applied electric and magnetic fields offer potential for novel devices. These materials are often complex, and thus present challenges for materials design. We address the role of the sign of the microscopic magnetic interaction for coupling to ferroelectric fluctuations and illustrate the surrounding issues with results in three different model systems, (Se,Te)CuO3, DyMnO3, and Ba0.5Sr1.5Zn2Fe12O22. We show from phase space considerations that antiferromagnetism exhibits a much larger coupling to the dielectric constant, and therefore to ferroelectricity, than ferromagnetism. We show in several materials how this symmetry requirement becomes a design principle. Specifically, an antiferromagnet can generate both the magnetoelectric coupling needed for ferroelectricity as well as a Zeeman coupling needed for field control. The latter arises from a parasitic canted FM state, opening up the possibility of magnetoelectric effects in broader classes of materials.