The fine structure of the heavy-hole exciton in CdSe quantum dots grown on ZnSe by molecular beam epitaxy is studied. The application of a tilted magnetic field allows us to observe all four levels of the ground state simultaneously. The zero-field separation between the optically forbidden and allowed doublets ranges between 1.7 and 1.9 meV and is markedly reduced due to the existence of a wetting layer and the finite energy barrier at the hetero-interface. The energy splitting within the allowed doublet (200 mu eV) is substantially larger than for the forbidden states (less than or equal to 20 mu eV), proving that a dominant contribution arises from the long-ranged part of the electron-hole exchange interaction. The total set of g-factors is derived. A huge anisotropy for the electron is found.