We report an unusual magnetic anisotropy in a S = 1/2 1D antiferromagnetic (AF) compound CuCl2(pdz) (pdz = pyridazine). The magnetic susceptibility for H//a* and H//c showed characteristic behavior in the S = 1/2 1D Heisenberg AF system, whereas that for H//b exhibited a 1/T contribution. The origin of such an anomalous anisotropy in the magnetic susceptibility is explained by the staggered g-tensor of this compound.