Abstract  The blazar 3C 454.3 exhibited a strong flare seen in γ-rays, X-rays and optical/near-infrared bands during 2009 December 3–12. Emission in the V and J bands rose more gradually than did the γ-rays and soft X-rays, though all peaked at nearly the same time. Optical polarization measurements showed dramatic changes during the flare, with a strong anticorrelation between optical flux and degree of polarization (which rose from ∼3 to ∼20 per cent) during the declining phase of the flare. The flare was accompanied by large rapid swings in polarization angle of ∼170°. This combination of behaviours appears to be unique. We have cm-band radio data during the same period but they show no correlation with variations at higher frequencies. Such peculiar behaviour may be explained using jet models incorporating fully relativistic effects with a dominant source region moving along a helical path or by a shock-in-jet model incorporating three-dimensional radiation transfer if there is a dominant helical magnetic field. We find that spectral energy distributions at different times during the flare can be fit using modified one-zone models where only the magnetic field strength and particle break frequencies and  normalizations need change. An optical spectrum taken at nearly the same time provides an estimate for the central black hole mass of ∼2.3 × 109 M⊙. We also consider two weaker flares seen during the ∼200 d span over which multiband data are available. In one of them, the V and J bands appear to lead the γ-ray and X-ray bands by a few days; in the other, all variations are simultaneous.