In recent years, the H-rotor vertical-axis turbine has attracted considerable attention in the field of wind and tidal power generation. After a series of complex spatiotemporal evolutions, the vortex shed from turbine blades forms a turbulent wake with a multi-scale coherent structure. An analysis of the wake characteristics of twin turbines forms the basis of array optimisation. This study aimed to examine the instability characteristics of a twin-turbine wake with two rotational configurations. The dynamic evolution characteristics of coherent structures with different scales in the wake were analysed via wavelet analysis. The results show that an inverse energy cascade process occurs after the high-frequency small-scale coherent structures induced by rotation lose their coherence. This self-organising characteristic is more apparent in the quasi two-dimensional wake of a forward-moving counter-rotating turbine (Array 1) than in that of a backward-moving counter-rotating turbine (Array 2). With greater organisation and coherence, the wake of Array 1 exhibits low-frequency instability characteristics dominated by a large-scale coherent structure. In addition, the signals reconstructed using wavelet transform show that asymmetric modes exist between low-frequency large-scale coherent structures. The experimental results provide a new perspective on the instability mechanism of twin-turbine wakes, as well as important data for numerical modelling.