  As a general framework for the studies of $t$-structures on triangulated
categories and torsion pairs in abelian categories, we introduce the notions of
extriangulated categories with negative first extensions and $s$-torsion pairs.
We define a heart of an interval in the poset of $s$-torsion pairs, which
naturally becomes an extriangulated category with a negative first extension.
This notion generalizes hearts of $t$-structures on triangulated categories and
hearts of twin torsion pairs in abelian categories. In this paper, we show that
an interval in the poset of $s$-torsion pairs is bijectively associated with
$s$-torsion pairs in the corresponding heart. This bijection unifies two
well-known bijections: One is the bijection induced by HRS-tilt of
$t$-structures on triangulated categories. The other is Asai--Pfeifer's and
Tattar's bijections for torsion pairs in an abelian category, which is related
to $\tau$-tilting reduction and brick labeling.
