Empirical tests for symmetry of cross-price elasticities It is a well known fact in consumer theory that for a Hicksian demand curve the cross-price elasticity of good $i$ with respect to the price of good $j$ equals the cross-price elasticity of good $j$ with respect to the price of good $i$. This result (iirc) depends only on the utility function being continuous and so has great generality.

Of course, no consumer is actually Hicksian (though firms of course are) but the vast majority of goods take up only a small portion of a person's expenditure and so we expect the income effect to be small, and so the consumer de-facto Hicksian. What is the empirical status of this prediction? Do economists agree with my claim that most consumers are de-facto Hicksians? If the answer to the second question is no, have similar empirical tests been done for firms?
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.