We prove a generalized version of the Lefschetz fixed point theorem, and use it to obtain a variety of periodic and aperiodic solutions for differential delay equations; in particular, of the type \dot x(t) = f(x(t-1)). Here f:\mathbb R \to \mathbb R is odd and two-periodic, and we obtain both strictly periodic solutions and solutions periodic modulo a multiple of two. The qualitative behavior of solutions can be coded by symbol sequences containing the sequence of levels about which these solutions oscillate.