On the archetypal `flavours', indices and teleconnections of ENSO revealed by global sea surface temperatures

El Ni\~no-Southern Oscillation global (ENSO) imprint on sea surface temperature comes in many guises. To identify its tropical fingerprints and impacts on the rest of the climate system, we propose a global approach based on archetypal analysis (AA), a pattern recognition method based on the identification of extreme configurations in the dataset under investigation. Relying on detrended sea surface temperature monthly anomalies over the 1982 to 2022 period, the technique recovers central and eastern Pacific ENSO types identified by more traditional methods and allows one to hierarchically add extra flavours and nuances to both persistent and transient phases of the phenomenon. Archetypal patterns found compare favorably to phase identification from K-means, fuzzy C-means and recently published network-based machine-learning algorithms. The AA implementation is modified for the identification of ENSO phases in sub-seasonal-to-seasonal prediction systems and complements current alert systems in characterising the diversity of ENSO and its teleconnections. Tropical and extra-tropical teleconnection composites from various oceanic and atmospheric fields derived from the analysis are shown to be robust and physically relevant. Extending AA to sub-surface ocean fields improves the discrimination between phases when the characterisation of ENSO based on sea surface temperature is uncertain. We show that AA on detrended sea-level monthly anomalies provides a clearer expression of ENSO types.