Empirical tests for symmetry of cross-price elasticities
Even if it is a very general result, its implications are not immediately intuitive and actually often quite difficult to accept. Why should the impact of an increase in the price for fresh strawberries on the demand for gasoline, be the same as the impact of a price increase for gasoline on the demand for fresh strawberries (while keeping $u$ constant) ?
One of the more convincing empirical investigation of Slutsky symmetry is experimental and provided in this paper which is also very closely related to the questions your ask:
Sippel, Reinhard, 1997, "An Experiment on the Pure Theory of Consumer's Behaviour," The Economic Journal, 107, 1431-1444.