Context. Gaia Data Release 3 (Gaia DR3) time series data may contain spurious signals related to the time-dependent scan angle. Aims. We aim to explain the origin of scan-angle-dependent signals and how they can lead to spurious periods, provide statistics to identify them in the data, and suggest how to deal with them in Gaia DR3 data and in future releases. Methods. Using real Gaia (DR3) data alongside numerical and analytical models, we visualise and explain the features observed in the data. Results. We demonstrated with Gaia (DR3) data that source structure (multiplicity or extendedness) or pollution from close-by bright objects can cause biases in the image parameter determination from which photometric, astrometric, and (indirectly) radial velocity time series are derived. These biases are a function of the time-dependent scan direction of the instrument and thus can introduce scan-angle-dependent signals, which due to the scanning-law-induced sampling of Gaia can result in specific spurious periodic signals. Numerical simulations in which a period search is performed on Gaia time series with a scan-angle-dependent signal qualitatively reproduce the general structure observed in the spurious period distribution of photometry and astrometry, and the associated spatial distributions on the sky. A variety of statistics allows for the deeper understanding and identification of affected sources. Conclusions. The origin of the scan-angle-dependent signals and subsequent spurious periods is well understood and is mostly caused by fixed-orientation optical pairs with a separation < 0.5″ (including binaries with P ≫ 5 y) and (cores of) distant galaxies. Although most of the sources with affected derived parameters have been filtered out from the Gaia archive nss_two_body_orbit and several vari-tables, Gaia DR3 data remain that should be treated with care (no sources were filtered from gaia_source). Finally, the various statistics discussed in the paper can be used to identify and filter affected sources and also reveal new information about them that is not available through other means, especially in terms of binarity on sub-arcsecond scale.