  We study the numbers of involutions and their relation to Frobenius-Schur
indicators in the groups $\mathrm{SO}^{\pm}(n,q)$ and $\Omega^{\pm}(n,q)$. Our
point of view for this study comes from two motivations. The first is the
conjecture that a finite simple group $G$ is strongly real (all elements are
conjugate to their inverses by an involution) if and only if it is totally
orthogonal (all Frobenius-Schur indicators are 1), and we are able to show this
holds for all finite simple groups $G$ other than the groups
$\mathrm{Sp}(2n,q)$ with $q$ even or $\Omega^{\pm}(4m,q)$ with $q$ even. We
prove computationally that for small $n$ and $m$ this statement indeed holds
for these groups by equating their character degree sums to the number of
involutions. We also prove a result on a certain twisted indicator for the
groups $\mathrm{SO}^{\pm}(4m+2,q)$ with $q$ odd. Our second motivation is to
continue the work of Fulman, Guralnick, and Stanton on generating function and
asymptotics for involutions in classical groups. We extend their work by
finding generating functions for the numbers of involutions in
$\mathrm{SO}^{\pm}(n,q)$ and $\Omega^{\pm}(n,q)$ for all $q$, and we use these
to compute the asymptotic behavior for the number of involutions in these
groups when $q$ is fixed and $n$ grows.
