High swirl number flows with vortex breakdown exhibit a number of unsteady flow features, including shear-induced coherent structures and precessing recirculation zones. This article analyzes how precession influences the relationship between the reacting flows' time-averaged and instantaneous features. Its objective is to provide interpretive insights into high fidelity computations or experimental results. It shows how precession influences three significant topological features in the time-averaged flow: (1) centerline axial jets, (2) centerline stagnation points, and (3) symmetry of the flow about the centerline. It also discusses the extent to which these first two features provide insight into the actual instantaneous flow topology. A particularly significant result of this work is in regards to aerodynamically stabilized flames. Stabilization of such flames requires a low velocity interior stagnation point(s), presumably in the vortex breakdown region. We show how precession causes systematic differences between the location of the time-averaged position of the instantaneous stagnation point, and the stagnation point of the time-averaged velocity. An important implication of this point is that a perfect prediction of the time-averaged flow field could still lead to a completely erroneous time-averaged flame position prediction. Finally, we discuss the influence of precession and coherent motion on convergence of estimated averaged quantities.