Let $G$ be an arbitrary simple graph of order $n$. We say that $G$ is a spectral complementary graph if $P_G(\lambda)-P_{\bar G}(\lambda)=(-1)^n(P_G(-\lambda-1)-P_{\bar G}(-\lambda-1))$. In this paper we investigate some properties of such graphs. In particular, we prove that a graph $G$ is spectral complementary if and only if its Seidel spectrum $\sigma^*(G)$ is symmetric with respect to the zero point. Besides, we determine all connected spectral complementary graphs of order $n=2,3,\dots,8$.