This paper extends the applicability of normal distributional theory in linear regression. In the classical linear model, the tstatistic follows the well-known Student tdistribution. Inspired by C. R. Rao’s characterization results, we give new conditions under which this result holds even if the errors are not i.i.d. normal. This is based on observing a novel symmetry: on considering two linear regressions, the direct regression (regression of yon xand Z) and the reverse regression (regression of xon yand Z), the tstatistic (tD) for the coefficient of xin the direct regression is numerically identical to the tstatistic for yin the reverse regression (regression of xon yand Z). This implies that their distributions are also identical. We show this yields a new type of condition under which a tstatistic follows the usual Student tdistribution, without an assumption on the conditional distribution of the dependent variable given the regressors. We extend these results to Fstatistics as well as various statistics in multivariate linear regressions. A simulation study confirms our theoretical results. We also present examples including field experiments, simultaneous equations models, and analysis of discrimination, to which our findings can be applied.