  We study the magnetically-induced phonon splitting in cubic ACr$_2$O$_4$
(A=Mg, Zn, Cd, Hg) spinels from first principles, and demonstrate that the sign
of the splitting, which is experimentally observed to be opposite in
CdCr$_2$O$_4$ compared to ZnCr$_2$O$_4$ and MgCr$_2$O$_4$, is determined solely
by the particular magnetic ordering pattern observed in these compounds. We
further show that this interaction between magnetism and phonon frequencies can
be fully described by the previously proposed spin-phonon coupling model that
includes only the nearest neighbor exchange. Finally, using this model with
materials specific parameters calculated from first principles, we provide
additional insights into the physics of spin-phonon coupling in this intriguing
family of compounds.
