Why is the HLR theory particle-hole symmetric?

Long wavelength descriptions of a half-filled lowest Landau level ($\nu = 1/2$) must be consistent with the experimental observation of particle-hole (PH) symmetry. The traditional description of the $\nu=1/2$ state pioneered by Halperin, Lee and Read (HLR) naively appears to break PH symmetry. However, recent studies have shown that the HLR theory with weak quenched disorder can exhibit an emergent PH symmetry. We find that such inhomogeneous configurations of the $\nu=1/2$ fluid, when described by HLR mean-field theory, are tuned to a topological phase transition between an integer quantum Hall state and an insulator of composite fermions with a dc Hall conductivity $\sigma_{xy}^{\rm (cf)} = - {1 \over 2} {e^2 \over h}$. Our observations help explain why the HLR theory exhibits PH symmetric dc response.