The complementary prism of a graph arises from the disjoint union of and the complement of by adding a perfect matching joining corresponding pairs of vertices in and . Partially answering a question posed by Haynes et al. (2007) we provide an efficient characterization of the circumference of the complementary prism of a tree and show that has cycles of all lengths between 3 and its circumference. Furthermore, we prove that for a given graph of bounded maximum degree it can be decided in polynomial time whether its complementary prism is Hamiltonian.