We study the effect of polydispersity on the compression and interpenetration properties of two opposing polymer brushes by numerical self-consistent field approach and by analytical theory. Polydispersity is represented by an experimentally relevant Schulz-Zimm chain-length distribution. We focus on three different polydispersities representing sharp, moderate, and extremely wide chain length distributions and derive approximate analytical expressions for the pressure-separation curves, Pi(D). We study the brush interpenetration and quantify it in terms of the overlap integral, Gamma, representing the number of interbrush contacts, and interpenetration length, delta. For the case of moderate densities where the equation of state is dominated by the second virial term with coefficient v, we demonstrate that the pressure, the overlap integral, and the interpenetration length are related by a simple equation, Pi/k(B)T = v Gamma/delta, where k(B)T represents the thermal energy. We propose a scaling form for delta(D) for the three polydispersity cases and compare these to the numerical results. A qualitative difference is observed in the behavior of the interpenetration length as a function of the compression distance: Whereas for monodisperse brushes delta scales as delta similar to D-1/3, with increasing polydispersity the slope of delta(D) becomes much smaller and eventually changes sign. This can be traced back to the change in the curvature of the brush density profile from convex up to convex down.