Nonparametric analysis of variance tests for factorial experiments, available in the literature, relate mostly to main effects in two-way layout linear models without interactions. Although the approach of Lehmann (1964) (see also Puri and Sen (1967)) can be adapted to construct tests for interactions, such tests are rather tedious to apply and are only asymptotically distribution-free. To obviate these drawbacks, the theory of permutationally distribution-free rank order tests for main effects, developed independently by the two authors, Sen (1968), Mehra (1968) (see also Mehra and Sarangi (1967)), is further extended in this paper to provide suitable tests for interactions in factorial experiments. The asymptotic power-efficiencies of the proposed tests are also studied. [The results of this paper were obtained independently by the two authors using, respectively, extensions of the Hajek (1962) and the Chernoff-Savage (1958) theorems on the asymptotic normality of rank order statistics.]