First remove seasonal trend or long-term trend in time series? I have a time series (quarterly data) which has both a long-term trend and seasonality. Taking seasonal differences will make the series stationary, according to the Augmented Dickey-Fuller test. On the other hand, if I first take non-seasonal differences, the series also become stationary according to the Augmented Dickey-Fuller test, but the ACF still shows seasonal correlation.
How do I tackle this? Should I take both seasonal -and non seasonal differences because there is a trend and seasonality, even though taking only of them already makes the series stationary?
I am confused because examples I found seem to be contradictory.
 A: As you suggested you one can observe non-stationarity (symptom) but the correct remedy (medicine) is unclear. The correct remedy could be multiple level shifts , multiple trends , seasonal pulses too name a few. Assuming any one approach is both simple and potentially damaging to good statistical analysis.. The high road is to to "listen to the data" ala Bacon,Box,Tukey et al and form the appropriate form of non-stationarity adjustment (much like a good drug prescription) to render the data stationary without incurring damage. The whole idea is to keep the model simple but not too simple.
Non-stationarity can be induced by changes in model form , changes in parameters, changes in error variance besides what has been previously listed here. The message is avoid presumptive cook-book rules suggested by some textbooks and commentators particularly x-11 and it's variants and use the best statistical tools available to form/identify usable solutions.
For example review the outliers and changing error variance when using x11 on the classic airline series to construct the irregular or error process.

A: Remove them simultaneously. This example shows you how to do it. Henderson filter extracts the trend while S(3,3) filter extracts the seasonality. In fact all de-seasonalizing software will do this, I think.
