Tests based on the linear signed rank statistics are proposed for testing the symmetry of a continuous distribution about a known median. Among the linear signed rank statistics, the Wilcoxon test has gotten much attention in the nonparametric literature. In this article we propose to use the scores such as the squares of the Wilcoxon scores and the normal scores. The proposed statistics are distribution-free under the null hypothesis. Critical values of the proposed test statistics at several selected levels for sample sizes n = 10(1)20, 25, 30 are obtained. Through Monte Carlo studies we compare the performance of our proposed tests with the Hill and Rao's statistic, McWilliams's runs statistic and Wilcoxon signed rank statistic. The results show that the proposed tests are more powerful than the competitive tests under a broad class of alternatives.