The wake of a streamwise oscillating circular cylinder has been experimentally investigated over a range of oscillation amplitude and frequency ratios using laser-induced-fluorescence flow visualization, particle image velocimetry and hot-wire techniques. Five typical flow structures, referred to as S-I, S-II, A-I, A-III and A-IV, are identified. Special attention is given to the S-II mode because this flow structure is observed experimentally for the first time. It consists of two rows of binary vortices symmetrically arranged about the centreline of the wake. Each binary vortex contains two counter-rotating vortices shed from the same side of the cylinder. This flow structure corresponds to zero mean and fluctuating lift on the cylinder, which could be of engineering significance. A theoretical analysis for this flow has been conducted based on the governing equations. The solution to the two-dimensional vorticity equation suggests that the flow may be considered to be the superposition of two components, i.e. that due to a stationary cylinder in a steady uniform cross-flow and to a cylinder oscillating in fluid at rest, which are characterized by alternate and symmetric vortex shedding, respectively. The solution provides insight into the formation of the various modes of the flow structure. A semi-empirical prediction of the S-II mode structure is developed, which is in excellent agreement with experimental data as well as with previous numerical results.