  We study $t$-structures (on triangulated categories) that are closely related
to weight structures.
  A $t$-structure couple $t=(C_{t\le 0},C_{t\ge 0})$ is said to be adjacent to
a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$.
  For a category $C$ that satisfies the Brown representability property we
prove that $t$ that is adjacent to $w$ exists if and only if $w$ is smashing
(that is, "respects C-coproducts"). The heart $Ht$ of this $t$ is the category
of those functors $Hw^{op}\to Ab$ that respect products (here $Hw$ is the heart
of $w$); the result has important applications.
  We prove several more statements on constructing $t$-structures starting from
weight structures; we look for a strictly orthogonal $t$-structure $t$ on some
$C'$ (where $C,C'$ are triangulated subcategories of a common $D$) such that
$C'_{t\le 0}$ (resp. $C'_{t\ge 0}$) is characterized by the vanishing of
morphisms from $C_{w\ge 1}$ (resp. $C_{w\le -1}$). Some of these results
generalize properties of semi-orthogonal decompositions proved in the previous
paper, and can be applied to various derived categories of (quasi)coherent
sheaves on a scheme $X$ that is projective over an affine noetherian one. We
also study hearts of orthogonal $t$-structures and their restrictions, and
prove some statements on "reconstructing" weight structures from orthogonal
$t$-structures.
