The aim of this paper is to gain new insight into the vector spaces of pencils \({\mathbf L}_1(P)\) and \({\mathbf L}_2(P)\), and their intersection \(\text{DL}(P)\), that arise in connection with the linearization of the polynomial eigenvalue problem \(P(\lambda)x = 0\). The authors show that for arbitrary polynomials every pencil in \(\text{DL}(P)\) is block symmetric and obtain a convenient basis for \(\text{DL}(P)\) built from block Hankel matrices. When \(P\) is symmetric it is shown that the symmetric pencils in \({\mathbf L}_1(P)\) comprise \(\text{DL}(P)\), while for Hermitian \(P\) the Hermitian pencils in \({\mathbf L}_1(P)\) form a proper subset of \(\text{DL}(P)\) that is explicitly characterized. In addition to obtaining new results, this paper provides a self-contained treatment of some of the key properties of \(\text{DL}(P)\) together with some new, more concise proofs.