The level crossing mechanism between the ground and the first excited state of Na:Fe-6 antiferromagnetically coupled iron rings is studied by torque magnetometry down to 40 mK and in magnetic fields up to 28 T. The step width at the crossing field B. assumes a finite value at the lowest temperatures. This fact is ascribed to the presence of level anticrossing, not expected for a ring with axial, i.e. S-6 point group, symmetry. Assuming a reduced symmetry, we revised the model Hamiltonian of such a spin system by introducing a Dzyaloshinsky-Moriya (DM) term and we show, by exact diagonalization, that DM term can account for the mixing of states with different parity. In particular, analytical as well numerical analysis show that the introduction of the DM term may contribute to the broadening of the torque step as well as for the finite energy gap at B, observed by heat capacity in a similar ring Li:Fe-6 as previously reported [9].