  In this paper, we describe a new infinite family of $\frac{q^{2}-1}{2}$-tight
sets in the hyperbolic quadrics $\mathcal{Q}^{+}(5,q)$, for $q \equiv 5 \mbox{
or } 9 \bmod{12}$. Under the Klein correspondence, these correspond to
Cameron--Liebler line classes of ${\rm PG}(3,q)$ having parameter
$\frac{q^{2}-1}{2}$. This is the second known infinite family of nontrivial
Cameron--Liebler line classes, the first family having been described by Bruen
and Drudge with parameter $\frac{q^{2}+1}{2}$ in ${\rm PG}(3,q)$ for all odd
$q$.
  The study of Cameron--Liebler line classes is closely related to the study of
symmetric tactical decompositions of ${\rm PG}(3,q)$ (those having the same
number of point classes as line classes). We show that our new examples occur
as line classes in such a tactical decomposition when $q \equiv 9 \bmod 12$ (so
$q = 3^{2e}$ for some positive integer $e$), providing an infinite family of
counterexamples to a conjecture made by Cameron and Liebler in 1982; the nature
of these decompositions allows us to also prove the existence of a set of type
$\left(\frac{1}{2}(3^{2e}-3^{e}), \frac{1}{2}(3^{2e}+3^{e}) \right)$ in the
affine plane ${\rm AG}(2,3^{2e})$ for all positive integers $e$. This proves a
conjecture made by Rodgers in his PhD thesis.
