We have studied the fine structure of the heavy-hole exciton in CdSe quantum dots grown on ZnSe by molecular beam epitaxy. Applying a tilted magnetic field, all four levels of the ground state are detected simultaneously. The zero-field separation between the optically forbidden and allowed doublets ranges between 1.7 and 1.9 meV and is crucially influenced by the finite energy barrier at the heterointerface as well as the existence of a wetting layer. The energy splitting within the allowed doublet (200 μeV) is substantially larger than for the forbidden states (⩽ 20 μeV), demonstrating 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.