  We report the experimental observation of spin reorientation in the double
perovskite Ho$_2$FeCoO$_6$. The magnetic phase transitions in this compound are
characterized and studied through magnetization and specific heat, and the
magnetic structures are elucidated by neutron powder diffraction. Two magnetic
phase transitions are observed in this compound - one at $T_\mathrm{N1}
\approx$ 250~K, from paramagnetic to antiferromagnetic, and the other at
$T_\mathrm{N2} \approx$ 45~K, from a phase with mixed magnetic structures to a
single phase through a spin reorientation process. The magnetic structure in
the temperature range 200~K - 45~K is a mixed phase of the irreducible
representations $\Gamma_1$ and $\Gamma_3$, both of which are antiferromagnetic.
The phase with mixed magnetic structures that exists in Ho$_2$FeCoO$_6$ gives
rise to a large thermal hysteresis in magnetization that extends from 200~K
down to the spin reorientation temperature. At $T_\mathrm{N2}$, the magnetic
structure transforms to $\Gamma_1$. Though long-range magnetic order is
established in the transition metal lattice, it is seen that only short-range
magnetic order prevails in Ho$^{3+}$ - lattice. Our results should motivate
further detailed studies on single crystals in order to explore spin
reorientation process, spin switching and the possibility of anisotropic
magnetic interactions giving rise to electric polarization in Ho$_2$FeCoO$_6$.
