When analysis business cycles, it is usual to factor out any seasonal influences from the data. Theoretically, this makes perfect sense if one can distinguish well long-term trends, fluctuations at business cycle frequencies and seasonal factors, both in separating them in the data and in the fact that they do not influence each other. In practice, the filtering is less than perfect, and there is at least some evidence that business cycles have an impact on trends.
Tommaso Proietti proposes a measure of the influence of seasonal cycles on the business cycle. Indeed, one does not observe the seasonally adjusted data, only an estimate of it. This means that there is some uncertainty about the true adjusted data, and thus any analysis of it should carry this uncertainty with it. This is of special importance when estimating the output gap, as it is of high policy relevance and it has been shown that policy makers needs to take into account the uncertainty about its measurement. The bandpass and HP filters both exhibit quite some uncertainty in the measurement of the cyclical components. This means in particular that one should carry the analysis from data that has not been seasonally adjusted, using a procedure suggested in the paper. I am not qualified to judge whether this is the best method, but this should at least make us more careful with the data.