  A $t$-structure $t=(C_{t\le 0},C_{t\ge 0})$ on a triangulated category $C$ is
right adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge
0}=C_{w\ge 0}$; then $t$ can be uniquely recovered from $w$ and vice versa. We
prove that if $C$ satisfies the Brown representability property then $t$ that
is adjacent to $w$ exists if and only if $w$ is smashing (i.e., coproducts
respect weight decompositions); then the heart $Ht$ is the category of those
functors $Hw^{op}\to Ab$ that respect products. The dual to this statement is
related to results of B. Keller and P. Nicolas. We also prove that an adjacent
$t$ exists whenever $w$ is a bounded weight structure on a saturated $R$-linear
category $C$ (for a noetherian ring $R$); for $C=D^{perf}(X)$, where the scheme
$X$ is regular and proper over $R$, this gives 1-to-1 correspondences between
bounded weights structures on $C$ and the classes of those bounded
$t$-structures on it such that $Ht$ has either enough projectives or
injectives. We generalize this existence statement to construct (under certain
assumptions) a $t$-structure $t$ on a triangulated category $C'$ such that $C$
and $C'$ are subcategories of a common triangulated category $D$ and $t$ is
right orthogonal to $w$. In particular, if $X$ is proper over $R$ but not
necessarily regular then one can take $C=D^{perf}(X)$, $C'=D^b_{coh}(X)$ or
$C'=D^-_{coh}(X)$, and $D=D_{qc}(X)$. We also study hearts of orthogonal
$t$-structures and their restrictions, and prove some statements on
"reconstructing" weight structures from orthogonal $t$-structures. The main
tool of this paper are virtual $t$-truncations of (cohomological) functors;
these are defined in terms of weight structures and "behave as if they come
from $t$-truncations" whether $t$ exists or not.
