The ground-stare properties of the tetranuclear Co2+ cluster [Co-4(H2O)(2)(PW9O34)(2)](10-) were investigated by combining specific heat, magnetic susceptibility, and magnetization measurements with a detailed inelastic neutron scattering (INS) study on a fully deuterated sample of K-10[Co-4(H2O)(2)(PW9O34)(2)]. 22H(2)O As a result of the single ion anisotropy of the octahedral Co2+, the appropriate exchange Hamiltonian of the Co-4 spin cluster is anisotropic. INS turns out to be essential for the determination of energy splittings in the ground state resulting from the coupling. Besides the energy pattern, INS provides information about the wave functions of the split ground-state components of the spin cluster. Based on the Hamiltonian (H) over cap(AN) = Sigma(i=x,y,z) -2J(i)(S(1)iS(3)i + S(1)iS(4)i + S(2)iS(3)i + S(2)iS(4)i) - 2J'(i)(S(1)iS(2)i) the following set of parameters is obtained: J(z) = 1.51 meV, J(x) = 0.70 meV, J(z)' = 0.46 meV, J(x)' = 0.44 meV, r = 1.6, where r = J(x)/J(y) = J(x)'/J(y)'. These parameters also reproduce the magnetic specific heat, susceptibility, and magnetization properties of the compound.