regression models with seasonal arima errors: a question

[newsagent]

Sometimes I observe that when a regression model with sarima errors contains a variable, say x1, the ARIMA function proposes first seasonal difference (D=1), whereas when the regression model with sarima errors contains another variable, say x2, the ARIMA function proposes zero seasonal difference (D=0). Why this is the case? The forecast variable is the same in both models

[robjhyndman]

The ARIMA model is applied to the residuals from the regression. These can obviously change with different predictors.