The structure of the ground state exciton of Cd(Se,Te) quantum dots embedded in ZnTe matrix is studied experimentally using photoluminescence spectroscopy and theoretically using k • p and configuration interaction methods. The experiments reveal a considerable reduction of fine-structure splitting energy of the exciton with increase of Se content in the dots. That effect is interpreted by theoretical calculations to originate due to the transition from spatially direct (type-I) to indirect (type-II) transition between electrons and holes in the dot induced by increase of Se. The theory results match those of the experiment very well. The theory identifies that the main mechanism causing elevated fine-structure energy in particular in type-I dots is due to the multipole expansion of the exchange interaction. Moreover, the theory reveals that for Se contents in the dot > 0.3, there exist also a confinement related to transition between type I and type II and which exhibits extraordinary properties, such as almost purely light hole character of exciton and toroidal shape of hole states which might be utilized for realization of the Aharonov-Bohm effect.