 Particle-hole equality of tetragonal extremal orbit areas

At half filling the nearest-neighbor band $E-E_0=-2t(\cos x+\cos y)-t\cos z$ has an electron contour at $z=\pi$ and a hole contour at $z=0$. Translation $(x,y)\mapsto (x+\pi ,y+\pi )$ maps the enclosed pockets into each other, proving equal regular Onsager frequencies for a field along $z$. Comparing an occupied slice with the complement of a hole orbit is not a comparison of two different enclosed planar orbits. Additional band terms or changed filling can lift this equality.