We propose a unified framework for testing various assumptions commonly made for covariance functions of stationary spatio-temporal random fields. The methodology is based on the asymptotic normality of space–time covariance estimators. We focus on tests for full symmetry and separability in this article, but our framework naturally covers testing for isotropy and Taylor's hypothesis. Our test successfully detects the asymmetric and nonseparable features in two sets of wind speed data. We perform simulation experiments to evaluate our test and conclude that our method is reliable and powerful for assessing common assumptions on space–time covariance functions.