  The spinel NiCr$_2$O$_4$ is known to show a ferrimagnetic transition at $T_c
= 70$~K, and magneto-structural transitions at $T_s = 30$~K and $T_o =<EMAIL_ADDRESS>We present a detailed magnetic and magnetocaloric effect (MCE $= -\Delta
S_{M}(T)$) study across these transitions. The $-\Delta S_{M}(T)$ shows a
positive anomaly at $T_c$, $T_s$, and $T_o$. In addition to these anomalies, we
report a new unreported feature at $T \approx 8.5$~K where $-\Delta S_{M}(T)$
shows a negative anomaly or the inverse MCE. An Arrot plot of the isothermal
magnetization data reveals important information about the nature of the
possible phases revealed in $-\Delta S_{M}(T)$. We have also made a scaling
analysis of the $-\Delta S_{M}(T)$ data around these transitions. This analysis
suggests that the transition at $T_c$ is a second-order Mean field like
transition, the transition at $T_s$ is not second order and is non-mean field
like, while the new transition at $T = 8.5$~K is non-mean field like but is
second order in nature. Our study demonstrates that magnetocaloric effect is
sensitive to magneto-structural changes in materials and can be used for the
identification of new phases and transitions.
