Slutsky symmetry and negative semidefiniteness are necessary and sufficient conditions for the rationality of demand functions. While the empirical implications of Slutsky negative semidefiniteness in repeated cross-sectional demand data are well understood, the empirical content of Slutsky symmetry remains largely unexplored. This paper takes an important first step toward addressing this gap. We demonstrate that the average Slutsky matrix is not identified and that its identified set always contains a symmetric matrix. A key implication of our findings is that the symmetry of the average Slutsky matrix is untestable, and consequently, individual Slutsky symmetry cannot be tested using the average Slutsky matrix.