  We classify tuples of (not necessarily commuting) isometries that admit von
Neumann-Wold decomposition. We introduce the notion of twisted isometries for
tuples of isometries and prove the existence of orthogonal decomposition for
such tuples. The former classification is partially inspired by a result that
was observed more than three decades ago by Gaspar and Suciu. And the latter
result generalizes Popovici's orthogonal decompositions for pairs of commuting
isometries to general tuples of twisted isometries which also includes the case
of tuples of commuting isometries. Our results unify all the known orthogonal
decomposition related results in the literature.
