The GLAST-AGILE Support Program (GASP) of the Whole Earth Blazar Telescope (WEBT) monitored BL Lacertae in 2008-2009 at radio, near-IR, and optical frequencies. During this period, high-energy observations were performed by XMM-Newton, Swift, and Fermi. We analyse these data with particular attention to the calibration of Swift UV data, and apply a helical jet model to interpret the source broad-band variability. The GASP-WEBT observations show an optical flare in 2008 February-March, and oscillations of several tenths of mag on a few-day time scale afterwards. The radio flux is only mildly variable. The UV data from both XMM-Newton and Swift seem to confirm a UV excess that is likely caused by thermal emission from the accretion disc. The X-ray data from XMM-Newton indicate a strongly concave spectrum, as well as moderate flux variability on an hour time scale. The Swift X-ray data reveal fast (interday) flux changes, not correlated with those observed at lower energies. We compare the spectral energy distribution (SED) corresponding to the 2008 low-brightness state, which was characterised by a synchrotron dominance, to the 1997 outburst state, where the inverse-Compton emission was prevailing. A fit with an inhomogeneous helical jet model suggests that two synchrotron components are at work with their self inverse-Compton emission. Most likely, they represent the radiation from two distinct emitting regions in the jet. We show that the difference between the source SEDs in 2008 and 1997 can be explained in terms of pure geometrical variations. The outburst state occurred when the jet-emitting regions were better aligned with the line of sight, producing an increase of the Doppler beaming factor. Our analysis demonstrates that the jet geometry can play an extremely important role in the BL Lacertae flux and spectral variability.
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