Consider $p(\geqq 2)$ treatments in an experiment involving paired comparisons. The $(i, j)$th pair yields $n_{ij}$ observations $X_{ijm}$ with distribution functions (df's) $F_{ij}(x) = F(x - \beta_i + \beta_j), (1 \leqq m \leqq n_{ij}; 1 \leqq i < j \leqq p), \mathbf{\beta} = (\beta_1, \cdots, \beta_p)'$ being the vector involving treatment effects. We assume that $F$ is symmetric about zero, i.e. $F(x) + F(-x) = 1$, for all real $x$. This is, for example, the situation, when one considers a replicated balanced incomplete block design with each block of size two under the usual assumption of additivity in an analysis of variance model. The null hypothesis to be tested is as follows: (1.1) $H_0': F_{ij}(x) = F(x)$, for all real $x$, and $1 \leqq i < j \leqq p,$ against all alternatives, which is equivalent to (1.2) $H_0: \mathbf{\beta} = \mathbf{0},$ against $\beta \neq \mathbf{0, 0}$ being a $p$-component column vector. A class of rank-order tests for the above problem was considered in Mehra and Puri (1967) and Puri and Sen (1969b), and asymptotic distributions of test statistics were obtained both under the null hypothesis, and under a sequence of alternatives converging to the null hypothesis at a suitable rate. However, any asymptotic optimality properties of the test procedures were not considered in either of the two papers. For subsequent notational convenience, we first pool the $n = \sum\sum_{i\leqq i i,