A systematic study using neutron diffraction and magnetic susceptibility is reported on Mn substituted ferrimagnetic inverse spinel Ti1-x Mn x Co2O4 in the temperature interval 1.6 K [Formula: see text] T [Formula: see text] 300 K. Our neutron diffraction study reveals cooperative distortions of the TO6 octahedra in the Ti1-x Mn x Co2O4 system for all the Jahn-Teller active ions T = Mn3+ , Ti3+ and Co3+ , having the electronic configurations 3d 1, 3d 4 and 3d 6, respectively which are confirmed by the x-ray photoelectron spectroscopy. Two specific compositions (x = 0.2 and 0.4) have been chosen in this study because these two systems show unique features such as; (i) noncollinear Yafet-Kittel type magnetic ordering, and (ii) weak tetragonal distortion with c/a < 1, in which the apical bond length d c (T B -O) is longer than the equatorial bond length d ab (T B -O) due to the splitting of the e g level of Mn3+ ions into [Formula: see text] and [Formula: see text]. For the composition x = 0.4, the distortion in the T B O6 octahedra is stronger as compared to x = 0.2 because of the higher content of trivalent Mn. Ferrimagnetic ordering in Ti0.6Mn0.4Co2O4 and Ti0.8Mn0.2Co2O4 sets in at 110.3 and 78.2 K, respectively due to the presence of unequal magnetic moments of cations, where Ti3+ , Mn3+ , and Co3+ occupy the octahedral, whereas, Co2+ sits in the tetrahedral site. For both compounds an additional 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 (MCP) at T COMP = 25.4 K in Ti0.8Mn0.2Co2O4 and a reentrant spin-glass behaviour in Ti0.6Mn0.4Co2O4 with a freezing temperature of ∼110.1 K. The MCP in this compound 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.