  Let $\mathbb{A}$ and $\mathbb{S}$ denote the double arrow of Alexandroff and
the Sorgenfrey line, respectively. We show that any homeomorphism
$h:^m\mathbb{A}\to^m\mathbb{A} $ is locally (outside of a nowhere dense set) a
product of monotone embeddings $h_i:J_i\subseteq \mathbb{A}\to\mathbb{A} (i\in
m)$ followed by a permutation of the coordinates. We also prove that the
symmetric products $\mathcal{F}_m(\mathbb{A})$ are not homogeneous for any
$m\geq 2$. This partially solves an open question of A. Arhangel'ski\v{i}. In
contrast, we show that symmetric product $\mathcal{F}_2(\mathbb{S})$ is
homogeneous.
