Let $\mathcal{S}$ be a finite generalized quadrangle of
order $(s,t)$, $s \ne 1 \ne t$. A spread is a set of $st + 1$
mutually non-concurrent lines of $\mathcal{S}$. A spread
$\mathbf{T}$ of $\mathcal{S}$ is called a spread of symmetry
if there is a group of automorphisms of $\mathcal{S}$ which
fixes $\mathbf{T}$ elementwise and which acts transitively on
the points of at least one (and hence every) line of
$\mathbf{T}$. From spreads of symmetry of generalized
quadrangles, there can be constructed near polygons, and new
spreads of symmetry would yield new near polygons. In this
paper, we focus on spreads of symmetry in generalized
quadrangles of order $(s,s^2)$. Many new characterizations of
the classical generalized quadrangle $\mathcal{Q}(5,q)$ which
arises from the orthogonal group $\mathbf{O}^{-}(6,q)$ will be
obtained. In particular, we show that a generalized quadrangle
$\mathcal{S}$ of order $(s,t)$, $s \ne 1 \ne t$, containing a
spread of symmetry {\bf T} is isomorphic to
$\mathcal{Q}(5,s)$, under any of the following conditions:
¶ (i) $\mathcal{S}$ contains a point which is incident
with at least three axes of symmetry (Theorem 6.4); ¶
(ii) $t = s^2$ with $s$ even and $\mathcal{S}$ has a center of
transitivity (Theorem 6.6); ¶ (iii) there exists a line
$L \not\in \mathbf{T}$ such that $\mathcal{S}$ is an EGQ with
base-line $L$ (Theorem 6.8).