  Let $\sigma$ be an endomorphism and $\delta$ an $\sigma$-derivation of a ring
$R$. In this paper, we show that if $R$ is $(\sigma,\delta)$-skew Armendariz
and $a\sigma(b)=0$ implies $ab=0$ for $a,b\in R$. Then $R$ is symmetric
(respectively, reversible) if and only if $R$ is $\sigma$-symmetric
(respectively, $\sigma$-reversible) if and only if $R[x;\sigma,\delta]$ is
symmetric (respectively, reversible). Moreover, we study on the relationship
between the Baerness, quasi-Baerness and p.q.-Baerness of a ring $R$ and these
of the Ore extension $R[x;\sigma,\delta]$. As a consequence we obtain a partial
generalization of \cite{hong/2000}.
