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.