In this paper, we study the existence of periodic solutions to the following delay differential system: {x(t)=∫f(x(t+θ))dθ,x(t)=∫f(x(t+θ))dθ. By transforming the problem of searching for periodic solutions of the above system to the problem of finding periodic solutions of an associated ordinary differential system with boundary value conditions and using pseudoindex theory, some sufficient conditions are obtained to guarantee the existence of multiple W-symmetric 2-periodic solutions. We also present three specific examples to illustrate our results.