We study the phonon-induced flip of the exciton spin in single strongly confining quantum dots. The considered two-phonon process contributes to the exciton spin relaxation (longitudinal relaxation time T1) within the radiative doublet of the exciton ground state. The respective effective matrix element is driven by an interplay of the short-range exchange interaction and the lattice deformation induced by acoustic phonons. The two-phonon process involves the participation of the dipole-forbidden dark states. The here considered relaxation channel is of paramount importance for symmetrical dots and may be several orders of magnitude faster than the previously studied transition between bright states in asymmetrical quantum dots. The calculated relaxation rates depend on the dot composition and shape, and decrease very strongly upon reduction in the dot size. For various individual dots belonging to a large quantum-dot ensemble the respective relaxation times may differ by a several orders of magnitude. For a typical ensemble of InAs/GaAs self-organized quantum dots at low temperatures, the numerical estimates range from few hundreds ... mehr of microseconds for the largest dots to few tens of nanoseconds for the smallest dots involved.