  We report a phenomenological magneto-structural model based on competing
free-energy terms that couple either polar or chiral distortions in cubic
quadruple perovskites, depending on the global direction of magnetic moments.
The model naturally explains why some compounds in this material system host
magnetically-induced ferroelectricity at low temperature, while others such as
CaMn$_3$(Cr$_3$Mn)O$_{12}$, which we characterise experimentally, do not.
Importantly, our results suggest a new approach towards developing an applied
multiferroic functionality, and can be generalised to other multi-sublattice
systems where the magnetic interaction between sublattices is prohibited by
spatial inversion.
