  The complex magnetic structures, spin-reorientation and associated exchange
interactions have been investigate in Er$_{0.5}$Dy$_{0.5}$FeO$_3$ using bulk
magnetization, neutron diffraction, specific heat measurements and density
functional theory calculations. The Fe$^{3+}$ spins order as G-type
antiferromagnet structure depicted by ${\Gamma}_{4}$($G_{x}$,$A_{y}$,$F_{z}$)
irreducible representation below 700K, similar to its end compounds. The bulk
magnetization data indicate occurrence of the spin-reorientation and rare-earth
magnetic ordering below $\sim$75 K and 10 K, respectively. The neutron
diffraction studies confirm an "incomplete" ${\Gamma}_{4}$${\rightarrow}$
${\Gamma}_{2}$($F_{x}$,$C_{y}$,$G_{z}$) spin-reorientation initiated $\leq$75
K. Although, the relative volume fraction of the two magnetic structures varies
with decreasing temperature, both co-exist even at 1.5 K. At 8 K,
Er$^{3+}$/Dy$^{3+}$ moments order as $c_{y}^R$ arrangement develop, which
gradually increases in intensity with decreasing temperature. At 2 K, magnetic
structure associated with $c_{z}^R$ arrangement of Er$^{3+}$/Dy$^{3+}$ moments
also appears. At 1.5 K the magnetic structure of Fe$^{3+}$ spins is represented
by a combination of ${\Gamma}_{2}$+${\Gamma}_{4}$+${\Gamma}_{1}$, while the
rare earth moments coexists as $c_{y}^R$ and $c_{z}^R$ corresponding to
${\Gamma}_{2}$ and ${\Gamma}_{1}$ representation, respectively. The observed
Schottky anomaly at 2.5 K suggests that the "rare-earth ordering" is induced by
polarization due to Fe$^{3+}$ spins. The Er$^{3+}$-Fe$^{3+}$ and
Er$^{3+}$-Dy$^{3+}$ exchange interactions, obtained from first principle
calculations, primarily cause the complicated spin-reorientation and $c_{y}^R$
rare-earth ordering, respectively, while the dipolar interactions between
rare-earth moments, result in the $c_{z}^R$ type rare-earth ordering at 2 K.
