Wakes formed behind bluff bodies frequently reveal complex patterns of coherent vortical structures, with emergence of streamwise spatial periodicity particularly in the mid-wake region. In some cases, the vortex positions also maintain symmetry about the wake centreline. For the case in which two pairs of vortices are generated per shedding cycle, thereby constituting the so-called ‘2P’ mode wake, assumptions of spatial periodicity and symmetry allow for development of a mathematically tractable model using the point-vortex approximation. Our previous work (Basu & Stremler, Phys. Fluids, vol. 27 (10), 2015, 103603) considered staggered 2P wake configurations with two glide-reflective pairs of vortices shed in each period. Here we investigate the dynamics of a spatially periodic point-vortex street consisting of two pairs of vortices arranged with reflective symmetry about the streamwise centreline. Because of the symmetry, it is possible to model the spatially periodic point-vortex dynamics as an integrable Hamiltonian system. For a particular choice of initial condition, the topological structure of the Hamiltonian level curves is determined by location in a circulation–impulse parameter space. These Hamiltonian level curves delineate multiple regimes of motion, with all vortex motions within one regime being qualitatively identical. This approach thus enables identification and a full classification of all possible vortex motions in this constrained system. There also exist a limited number of equilibrium configurations with no relative vortex motion; some of these relative equilibria are neutrally stable to (appropriate) perturbations. Only one such neutrally stable equilibrium configuration continues to preserve the distinct four-vortex array, and numerical experiments indicate that these configurations are also neutrally stable to small perturbations that break the spatial symmetry. We apply this analysis to identify the parameter values necessary for co-existence of two closely spaced, neutrally stable Kármán vortex streets that preserve the assumed symmetry. Finally, comparison of the model dynamics to a wake pattern reported in the literature suggests that the classification of exotic wakes should be based on more details than just the number of vortices periodically shed by the body.