  A {\it symmetric companion matrix} is a matrix of the form $A +A^T$ where $A$
is a companion matrix all of whose entries are in $\{0,1\}$ and $A^T$ is the
transpose of $A.$ In this paper, we find the total number of primitive and the
total number of imprimitive symmetric companion matrices. We establish formulas
to compute the exponent of every primitive symmetric companion matrix. Hence
the exponent set for the class of primitive symmetric companion matrices is
completely characterized. We also obtain the number of primitive symmetric
companion matrices with a given exponent for certain cases.
