The magnetic ground state of biatomic Fe chains on the reconstructed (5 x 1)-Ir(001) surface is a cycloidal 120 degrees spin spiral. Spin-resolved scanning tunneling microscopy reveals a striking variation of magnetic field dependences among the chains, which we attribute to parity effects resulting from finite lengths. Numerical simulations show that the chains are divided in three symmetry classes with the exact number of atoms in the chain determining the size and direction of their net magnetic moment. In contrast to antiferromagnetic systems, the three-atom periodicity causes the effective anisotropy to alternate between out of plane, in plane, and quenched.