We consider two scale-free tests of normality based on the characterization of symmetric normal law by Ahsanullah, Kibria and Shakil (2014). Both tests have the $U$-empirical structure, but the first one is of integral type, while the second one is of Kolmogorov type. We discuss the limiting behavior of the test statistics and calculate their local exact Bahadur efficiency for location, skew and contamination alternatives.