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 $T_o = 20$~K@. 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.
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