A standard way of treating the polynomial eigenvalue problem $P(\lambda)x = 0$ is to convert it into an equivalent matrix pencil—a process known as linearization. Two vector spaces of pencils $\mathbb{L}_1(P)$ and $\mathbb{L}_2(P)$, and their intersection $\mathbb{DL}(P)$, have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which their constituent pencils inherit structure from $P$. For arbitrary polynomials we show that every pencil in $\mathbb{DL}(P)$ is block symmetric and we obtain a convenient basis for $\mathbb{DL}(P)$ built from block Hankel matrices. This basis is then exploited to prove that the first $\deg(P)$ pencils in a sequence constructed by Lancaster in the 1960s generate $\mathbb{DL}(P)$. When $P$ is symmetric, we show that the symmetric pencils in $\mathbb{L}_1(P)$ comprise $\mathbb{DL}(P)$, while for Hermitian $P$ the Hermitian pencils in $\mathbb{L}_1(P)$ form a proper subset of $\mathbb{DL}(P)$ that we explicitly characterize. Almost all pencils in each of these subsets are shown to be linearizations. In addition to obtaining new results, this work provides a self-contained treatment of some of the key properties of $\mathbb{DL}(P)$ together with some new, more concise proofs.