  The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing
the pairwise distances between vertices, and the distance Laplacian matrix is
$\mathcal{D}^L(G)=T(G)-\mathcal{D}(G)$, where $T(G)$ is the diagonal matrix of
row sums of $\mathcal{D}(G)$. We establish several general methods for
producing $\mathcal{D}^L$-cospectral graphs that can be used to construct
infinite families. We provide examples showing that various properties are not
preserved by $\mathcal{D}^L$-cospectrality, including examples of
$\mathcal{D}^L$-cospectral strongly regular and circulant graphs. We establish
that the absolute values of coefficients of the distance Laplacian
characteristic polynomial are decreasing, i.e., $|\delta^L_{1}|\geq \dots \geq
|\delta^L_{n}|$ where $\delta^L_{k}$ is the coefficient of $x^k$.
