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The connections between European rainfall variability in winter (from January to March) and Northern Hemisphere circulation patterns, as represented by 500-hPa geopotential height, are investigated in the period 1900-2014. Initially, three statistically significant pairs of linearly-related circulation and precipitation patterns, explaining 45% of the variance in the latter field, are identified. The first two essentially represent the North Atlantic Oscillation (NAO) and the East Atlantic (EA) patterns, while the third corresponds to a structure described in recent literature as a blend of the Pacific North America (PNA) pattern and the so-called Asia Bering North America (ABNA) pattern. The residual precipitation field is then examined for patterns that can introduce non-linear modulations into the circulation-precipitation links. It is so found that the NAO impact on European rainfall is modulated by North American and Pacific factors controlling cyclogenesis over Newfoundland. The positive-phase EA rainfall anomalies over Central Europe and the British Isles seem to be markedly affected by the NAO phase. Finally, a possible signature from the Circumglobal Teleconnection Pattern disrupting the NAO influence on East Mediterranean precipitation anomalies is detected.