Atmospheric circulation plays a key role in driving extreme weather events. A coupling measure, termed the co-recurrence ratio (α), has been introduced to study the link between surface extremes and large-scale circulation patterns. Stemming from dynamical systems theory, the co-recurrence ratio quantifies joint recurrences of multiple atmospheric variables. Here, we evaluate the robustness of the performance of the co-recurrence ratio to the choice of geographical domains, the variables used to characterize the large-scale circulation, and the thresholds used to define extremes. We adopt a bivariate focus, diagnosing the coupling between large-scale circulation patterns and European winter extremes in temperature, wind, and precipitation. We find that the choice of domain significantly affects the interpretation of the spatial patterns and how strongly the hazard variables are coupled to the circulation variables. For some of the extremes, there are also notable differences in the coupling depending on whether the atmospheric circulation is diagnosed at the surface, by sea-level pressure, or in the mid-troposphere, by 500 hPa geopotential height. Finally, changing the threshold values used to define the extremes does not affect the robustness of our coupling results for the extreme event types that we consider. We conclude that the co-recurrence ratio provides valuable insights into the atmospheric drivers of surface extremes, but our findings also highlight the importance of choosing key parameters in the analysis with care.