Complete the description of how the two auxiliary fields parallel one another.

In a dielectric the displacement field hides the bound charge so that Gauss’s law counts only free charge. In a magnetic material the auxiliary H field plays the same role, hiding the bound currents so that Ampère’s law counts only free current.

$\mathbf{D}$ removes bound charge from Gauss’s law and $\mathbf{H}$ removes bound current from Ampère’s law. Both are bookkeeping devices, and in both cases the force is still exerted by the original field.