  Spin reorientation and magnetisation reversal are two important features of
the rare-earth orthorhombic provskites ($RM$O$_{3}$'s) that have attracted a
lot of attention, though their exact microscopic origin has eluded researchers.
Here, using density functional theory and classical atomistic spin dynamics we
build a general Heisenberg magnetic model that allows to explore the whole
phase diagram of the chromite and ferrite compounds and to scrutinize the
microscopic mechanism responsible for spin reorientations and magnetisation
reversals. We show that the occurrence of a magnetization reversal transition
depends on the relative strength and sign of two interactions between
rare-earth and transition-metal atoms: superexchange and Dzyaloshinsky-Moriya.
We also conclude that the presence of a smooth spin reorientation transition
between the so-called $\Gamma_4$ and the $\Gamma_2$ phases through a coexisting
region, and the temperature range in which it occurs, depends on subtle balance
of metal--metal (superexchange and Dzyaloshinsky-Moriya) and metal--rare-earth
(Dzyaloshinsky-Moriya) couplings. In particular, we show that the intermediate
coexistence region occurs because the spin sublattices rotate at different
rates.
