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$_6$.
