  The eccentricity matrix $\mathcal E(G)$ of a connected graph $G$ is obtained
from the distance matrix of $G$ by leaving unchanged the largest nonzero
entries in each row and each column, and replacing the remaining ones with
zeros. In this paper, we consider the set $\mathcal C \mathcal T$ of clique
trees whose blocks have at most two cut-vertices \textcolor{blue}{of the clique
tree}. After proving the irreducibility of the eccentricity matrix of a clique
tree in $\mathcal C \mathcal T$ and finding its inertia indices, we show that
every graph in $\mathcal C \mathcal T$ with more than $4$ vertices and odd
diameter has two positive and two negative $\mathcal E$-eigenvalues. Positive
$\mathcal E$-eigenvalues and negative $\mathcal E$-eigenvalues turn out to be
equal in number even for graphs in $\mathcal C \mathcal T$ with even diameter;
that shared cardinality also counts the \textcolor{blue}{`diametrally
distinguished'} vertices. Finally, we prove that the spectrum of the
eccentricity matrix of a clique tree $G$ in $\mathcal C \mathcal T$ is
symmetric with respect to the origin if and only if $G$ has an odd diameter and
exactly two adjacent central vertices.
