  An order-preserving square in a string is a fragment of the form $uv$ where
$u\neq v$ and $u$ is order-isomorphic to $v$. We show that a string $w$ of
length $n$ over an alphabet of size $\sigma$ contains $\mathcal{O}(\sigma n)$
order-preserving squares that are distinct as words. This improves the upper
bound of $\mathcal{O}(\sigma^{2}n)$ by Kociumaka, Radoszewski, Rytter, and
Wale\'n [TCS 2016]. Further, for every $\sigma$ and $n$ we exhibit a string
with $\Omega(\sigma n)$ order-preserving squares that are distinct as words,
thus establishing that our upper bound is asymptotically tight. Finally, we
design an $\mathcal{O}(\sigma n)$ time algorithm that outputs all
order-preserving squares that occur in a given string and are distinct as
words. By our lower bound, this is optimal in the worst case.
