The skew symmetric Laplace uniform distribution SSLUD(μ) is introduced in Lohot, R. K. and Dixit, V. U. (2024) using the skewing mechanism of Azzalini (1985). Here we derive the most powerful (MP) test for symmetry of the SSLUD(μ). Since the form of the test statistic is complicated and it is difficult to obtain its exact distribution, critical values and the power of MP test are obtained using simulation. Further, we construct a confidence interval (CI) for parameter μ assuming asymptotic normality and empirical distribution of the maximum likelihood estimator of μ. These two methods are compared based on the average length and coverage probability of the CI. Finally, the CI of the parameter μ is constructed using data on the transformed daily percentage change in the price of NIFTY 50, an Indian stock market index given in Lohot, R. K. and Dixit, V. U. (2024).