  Given two $\mathbb{P}$-objects in some algebraic triangulated category, we
investigate the possible relations among the associated $\mathbb{P}$-twists.
The main result is that, under certain technical assumptions, the
$\mathbb{P}$-twists commute if and only if the $\mathbb{P}$-objects are
orthogonal. Otherwise, there are no relations at all. In particular, this
applies to most of the known pairs of $\mathbb{P}$-objects on hyperk\"ahler
varieties. In order to show this, we relate $\mathbb{P}$-twists to spherical
twists and apply known results about the absence of relations between pairs of
spherical twists.
