Context. Blazars are beamed active galactic nuclei (AGNs) known for their strong multi-wavelength variability on timescales ranging from years down to minutes. Many different models have been proposed to explain this variability. Aims. We aim to investigate the suitability of the twisting jet model presented in previous works to explain the multi-wavelength behaviour of BL Lacertae, the prototype of one of the blazar classes. According to this model, the jet is inhomogeneous, curved, and twisting, and the long-term variability is due to changes in the Doppler factor due to variations in the orientation of the jet-emitting regions. Methods. We analysed optical data of the source obtained during monitoring campaigns organised by the Whole Earth Blazar Telescope (WEBT) in 2019–2022, together with radio data from the WEBT and other teams, and γ-ray data from the Fermi satellite. In this period, BL Lacertae underwent an extraordinary activity phase, reaching its historical optical and γ-ray brightness maxima. Results. The application of the twisting jet model to the source light curves allows us to infer the wiggling motion of the optical, radio, and γ-ray jet-emitting regions. The optical-radio correlation shows that the changes in the radio viewing angle follow those in the optical viewing angle by about 120 days, and it suggests that the jet is composed of plasma filaments, which is in agreement with some radio high-resolution observations of other sources. The γ-ray emitting region is found to be co-spatial with the optical one, and the analysis of the γ-optical correlation is consistent with both the geometric interpretation and a synchrotron self-Compton (SSC) origin of the high-energy photons. Conclusions. We propose a geometric scenario where the jet is made up of a pair of emitting plasma filaments in a sort of double-helix curved rotating structure, whose wiggling motion produces changes in the Doppler beaming and can thus explain the observed multi-wavelength long-term variability.