The hyperfine parameters, obtained from an electron-spin-resonance study, of the laser-active Tl0(1) center in four alkali-halides have been analysed and compared to the Tl0(2) center results. It is shown that the anisotropic contribution to the hyperfine components is virtually the same for all the Tl0 centers but that the isotropic contribution varies widely in size and sign. This is attributed to the varying amount of s-mixing caused by the varying odd crystal-field component throughout the alkali-halides for the Tl0(1) center. For Tl0(2) such s-mixing does not occur because this center possesses inversion symmetry. The results obtained show that an analysis of the hyperfine components provides a powerful tool to determine whether or not a given np1 atom or ion center is positioned in a crystal field possessing inversion symmetry. As an illustration the hyperfine parameters of some Pb+(6p1) centers in KCl and RbCl are analyzed. © 1985.