We present four epochs of high-resolution infrared spectroscopy of the peculiar X-ray binary Cygnus X-3. The observations cover quiescent, small-flaring and outburst states of the system as defined by radio and X-ray monitoring. The underlying infrared spectrum of the source, as observed during radio and X-ray quiescence and small-flaring states, is one of broad, weak Heii and Nv emission. Spectral variability in this state is dominated by modulation at the 4.8-h orbital period of the system. H-band spectra confirm the significant hydrogen depletion of the mass donor. The closest spectral match to the quiescent infrared spectrum of Cyg X-3 is an early-type WN Wolf-Rayet star. In outburst, the infrared spectrum is dramatically different, with the appearance of very strong twin-peaked Hei emission displaying both day-to-day variability and V(iolet)/R(ed) variations with orbital phase. We argue that the twin-peaked emission cannot arise in relativistic jets or, unless the distance to Cyg X-3 is severely overestimated, an accretion disc. The most likely explanation appears to be an enhanced stellar wind from the companion. Thus X-ray and radio outbursts in this system are likely to originate in mass-transfer, and not disc, instabilities, and the lengthening of the orbital period will not be smooth but will be accelerated during these outbursts. Furthermore, the appearance of these lines is suggestive of an asymmetric emitting region. We propose that the wind in Cyg X-3 is significantly flattened in the plane of the binary orbit. This may explain the observed twin-peaked Hei features as well as reconciling a massive Wolf-Rayet secondary with the relatively small optical depth to X-rays, if the disc wind is inclined at some angle to the line of sight. A small set of observations following outburst, when the system was returning to a more quiescent X-ray and radio state, reveal strong Hei 2.058-μm absorption with a clear P Cygni profile, at the same time as the more common weak Heii and Nv features. In a disc-wind geometry this can be interpreted as absorption in the densest, accelerating regions of the wind which can be viewed directly if the disc is inclined at some angle to the line of sight.