electromagnetism - Are there Optical Magnetic Mirrors (OMMs) which reflect via interaction with the magnetic field?

The most familiar mirror relies on reflection from a metal, where the very strong interaction of the electric field component of the EM wave with the conduction electrons produces a reflected wave.

Dielectric mirrors rely on the weaker interaction of the electric field with bound electrons, and a periodic modulation of the dielectric constant (alternating quarter wave thicknesses) builds up a reflected wave with high efficiency.

Both of those reflect due to the interaction of the electric field with electrons - be they bound or free.

However, the paper Optical Magnetic Mirrors (Schwanecke et al. 2007 J. Opt. A: Pure Appl. Opt. 9 L1) begins with:

Abstract: We report the first demonstration of an optical magnetic mirror achieved by nanostructuring a metal surface. It reverses the magnetic field of an incident wave upon reflection, acting as an ‘optical frequency superconductor’.

The introduction goes on to say:

A magnetic mirror imposes extremely unusual electromagnetic boundary conditions: it does not reverse the electric field of a light wave upon reflection, but reverses the magnetic field. This property renders it distinctly different from a normal mirror, which instead reverses the electric field of a reflected wave...

And so the working definition of a magnetic mirror here seems to be a surface which produces a reflection without reversing the direction of the electric field, but instead reversing the direction of the magnetic field.

However it's not clear to me if this is due to the interaction with the magnetic field directly.

In the paper Optical Magnetic Mirrors without Metals (also here Liu et al. 2014, Vol. 1, No. 4, 250) a nano-patterned purely dielectric surface is said to be an Optical Magnetic Mirror (OMM). I understand that the effect is due resonant modes within the dielectric patterns, and some of these modes are called 'electric' and some 'magnetic' as a naming convention, somewhat like there are TE and TM modes in a dielectric fiber waveguide. However, aren't these still due to the interaction of the electric field?

Question: Are there surfaces, or relatively thin nano-structures which reflect light via interaction with the magnetic field of the wave? Is there a physics distinction between that magnetic interaction, and a patterned surface that simply passes the test of not inverting the electric field?