  Let $R \to S$ be a cyclically pure map of rings essentially of finite type
over the complex numbers $\mathbb{C}$. In this paper, we show that if $S$ has
Du Bois singularities, then $R$ has Du Bois singularities. Our result is new
even when $R \to S$ is faithfully flat. As a consequence, we show that under
the same hypotheses on $R \to S$, if $S$ has log canonical type singularities
and $K_R$ is Cartier, then $R$ has log canonical singularities.
