  We focus on $\Theta$-rich and almost $\Theta$-rich words over a finite
alphabet $\mathcal{A}$, where $\Theta$ is an involutive antimorphism over
$\mathcal{A}^*$. We show that any recurrent almost $\Theta$-rich word $\uu$ is
an image of a recurrent $\Theta'$-rich word under a suitable morphism, where
$\Theta'$ is again an involutive antimorphism. Moreover, if the word $\uu$ is
uniformly recurrent, we show that $\Theta'$ can be set to the reversal mapping.
We also treat one special case of almost $\Theta$-rich words. We show that
every $\Theta$-standard words with seed is an image of an Arnoux-Rauzy word.
