It has recently been found that an InAs/GaSb based electron-hole system exhibits insulating behavior with unusual properties when the numbers of occupied electron and hole Landau levels are equal [R. J. Nicholas, K. Takashina, M. Lakrimi, B. Kardynal, S. Khym, N. J. Mason, D. M. Symons, D. K. Maude, and J. C. Portal, Phys. Rev. Lett. 85, 2364 (2000)]. In this insulating state, the Hall resistance becomes symmetric (even) under field reversal [Rxy(B)=Rxy(-B)], and both the Hall and longitudinal resistances display reproducible fluctuations. In this paper we present experimental studies of the geometry dependence of this phenomenon. In particular, we show conclusively that the conduction responsible for the reproducible fluctuations and the symmetric Hall resistance occur due to the presence of the mesa edge. Further investigations of the edge conductance are presented. We show that as a function of magnetic field, the edge conduction shows a qualitatively opposite behavior to the conductivity in the sample interior. This is confirmed through measurements with an in-plane component of magnetic field. Also, the size of the conductance fluctuations is found to have a monotonic relationship with the absolute value of the conductance. A model based on counterpropagating edge channels is presented which qualitatively accounts for the observed behavior.