Sensitivity of the anatase and rutile phases of titanium dioxide to Swift Heavy Ion (SHI) irradiation was experimentally probed and compared with the predictions of the Coulomb explosion, analytical and inelastic thermal spike models of ion-matter interaction. Conforming to the predictions of all these models, our study indicated higher sensitivity of anatase to these ions than the rutile phase. A detailed examination however revealed that Coulomb explosion model cannot explain either the nature of variation of the interaction cross section of SHI with the energy deposited by these ions, Se to the target electrons, or the relative values of the threshold electronic energy loss, Seth of anatase and rutile. The analytical thermal spike (a-TS) model, using the available physicochemical data for this oxide, predicted that tracks cannot form either in anatase or in rutile by 297 MeV and 511 MeV Ni ions, while inelastic thermal spike (i-TS) model predicted formation of ion tracks by 297 MeV Ni ions and their absence with 511 MeV Ni ions in both anatase and rutile. Our observation agreed with the predictions of i-TS model albeit with a difference in the radius of the tracks. In addition, we observed halo of defect ridden crystalline region of much larger radius around the ion track. Interestingly, the radius of the halo scales with the velocity of the ions, which is opposite to the conventionally observed velocity effect.