Statistical signal processing algorithms often rely upon Gaussianity and time-reversibility, two important notions related to the probability structure of stationary random signals and their symmetry. Parametric models obtained via second-order statistics (SOS) are appropriate when the available data is Gaussian and time-reversible. On the other hand, evidence of nonlinearity, non-Gaussianity, or time-irreversibility favors the use of higher-order statistics (HOS). In order to validate Gaussianity and time-reversibility, and quantify the tradeoffs between SOS and HOS, consistent, time-domain chi-squared statistical tests are developed in this paper. Exact asymptotic distributions are derived to estimate the power of the tests, including a covariance expression for fourth-order sample cumulants. A modification of existing linearity tests in the presence of additive Gaussian noise is discussed briefly. The novel Gaussianity statistic is computationally attractive, leads to a constant-false-alarm-rate test and is well suited for parametric modeling because it employs the minimal HOS lags which uniquely characterize ARMA processes. Simulations include comparisons with an existing frequency-domain approach and an application to real seismic data. Time-reversibility tests are also derived and their performance is analyzed both theoretically and experimentally.