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Detailed investigations of time series features across climates, continents and variable types can progress our understanding and modelling ability of the Earth's hydroclimate and its dynamics. They can also improve our comprehension of the climate classification systems appearing in their core. Still, such investigations for seasonal hydroclimatic temporal dependence, variability and change are currently missing from the literature. Herein, we propose and apply at the global scale a methodological framework for filling this specific gap. We analyse over 13 000 earth-observed quarterly temperature, precipitation and river flow time series. We adopt the Koppen-Geiger climate classification system and define continental-scale geographical regions for conducting upon them seasonal hydroclimatic feature summaries. The analyses rely on three sample autocorrelation features, a temporal variation feature, a spectral entropy feature, a Hurst feature, a trend strength feature and a seasonality strength feature. We find notable differences to characterize the magnitudes of these features across the various Koppen-Geiger climate classes, as well as between continental-scale geographical regions. We, therefore, deem that the consideration of the comparative summaries could be beneficial in water resources engineering contexts. Lastly, we apply explainable machine learning to compare the investigated features with respect to how informative they are in distinguishing either the main Koppen-Geiger climates or the continental-scale regions. In this regard, the sample autocorrelation, temporal variation and seasonality strength features are found to be more informative than the spectral entropy, Hurst and trend strength features at the seasonal time scale.