A systematic study using neutron diffraction and magnetic susceptibility are reported on Mn substituted ferrimagnetic inverse spinel Ti$_{1-x}$Mn$_{x}$Co$_2$O$_4$ in the temperature interval 1.6 K $leq$ $T$ $leq$ 300 K. Our neutron diffraction study reveals cooperative distortions of the $T$O$_6$ octahedral for all the Jahn-Teller active ions $T$ = Mn$^{3+}$, Ti$^{3+}$ and Co$^{3+}$, which are confirmed by the X-ray photoelectron spectroscopy. Two specific compositions ($x$ = 0.2 and 0.4) have been chosen because of their unique features: noncollinear Yafet-Kittel type ordering, and weak tetragonal distortion with ${c/a}$ $<$ 1, in which the apical bond length $d_c$($T_B$-O) is longer than the equatorial $d_{ab}$($T_B$-O) due to the splitting of the $e_g$ level of Mn$^{3+}$ ions into $d_{x^2-y^2}$ and $d_{z^2}$. For $x$ = 0.4, the distortion in the $T_B$O$_6$ octahedra is stronger as compared to $x$ = 0.2 because of the higher content of trivalent Mn. Ferrimagnetic ordering in $x$ = 0.4 and $x$ = 0.2 sets in at 110.3 and 78.2 K, respectively due to the unequal magnetic moments of cations, where Ti$^{3+}$, Mn$^{3+}$, and Co$^{3+}$ occupying the octahedral, whereas, Co$^{2+}$ sits in the tetrahedral site. In addition, weak antiferromagnetic component could be observed lying perpendicular to the ferrimagnetic component. The analysis of static and dynamic magnetic susceptibilities combined with the heat-capacity data reveals a magnetic compensation phenomenon at $T_{COMP}$ = 25.4 K in $x$ = 0.2 and a reentrant spin-glass behaviour in $x$ = 0.4 with a freezing temperature $sim$110.1 K. The compensation phenomenon is characterized by sign reversal of magnetization and bipolar exchange bias effect below $T_{COMP}$ with its magnitude depending on the direction of external magnetic field and the cooling protocol.
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