  The mixed rare-earth orthochromite Nd$_{0.5}$Dy$_{0.5}$CrO$_{3}$ has a N\'eel
temperature ($T_\mathrm{{N}}$) of ${\sim}$ 175\,K, resulting in the G-type
antiferromagnetic ordering of Cr$^{3+}$ spins. The inverse susceptibility shows
a deviation from Curie-Weiss law at 230\,K, with a large effective paramagnetic
moment of 8.8\,${\mu}_{\mathrm{B}}$. The ZFC-FC magnetization bifurcate just
above $T_\mathrm{{N}}$ and show a distinct signature of spin reorientation near
60\,K. Neutron diffraction show that below $T_\mathrm{{N}}$, the Cr$^{3+}$
spins align in ${\Gamma}_{2}$ representation as ($F_{x}$, $G_{z}$). Below
60\,K, due to spin reorientation, the magnetic structure is in ${\Gamma}_{1}$
($G_{y}$) configuration. The neutron diffraction does not show any signature of
rare-earth ordering even at 1.5\,K. First principles density functional theory
calculations within GGA+U and GGA+U+SO approximations reveal that the G-type
antiferromagnetic order is the ground state magnetic structure of Cr sublattice
and the spin-reorientation of Cr$^{3+}$ spins can happen in the absence of
3d-4f interactions unlike in the case of orthoferrites. The specific heat shows
a `${\lambda}$' anomaly at $T_\mathrm{{N}}$, while at low temperature two
distinct Schottky anomalies are observed; a Schottky peak at 2\,K and an
additional step-like feature above 10\,K. Above $T_\mathrm{{N}}$, the magnetic
transition is preceded by structural anomalies as seen in our x-ray diffraction
and Raman measurements. The deviation of structural parameters near N\'eel
temperature is smaller. The phonon frequencies show deviation from the standard
anharmonic behaviour: first near 250\,K, due to magneto-volume effects while
the second deviation occurs near 200\,K due to spin-phonon coupling.
