  For a flat morphism $\pi \colon X \to T$ between smooth quasi-projective
varieties and its fiber $X_0$, we prove that spherical objects on $D^b(X)$
pushed-forward from $D^b(X_0)$ induce autoequivalences of $D^b(X_0)$ itself.
Our construction provides new derived symmetries for some singular varieties,
which include singular fibers of elliptic surfaces (commonly referred to as
Kodaira fibers) and type $III$ degenerations of K3 surfaces. In the case of
Kodaira fibers of type $I_n$, we also show the induced autoequivalences of
$D^b(X_0)$ correspond to the half twists on the $n$-punctured $2$-torus via
homological mirror symmetry. As an application, we describe the autoequivalence
groups of elliptic surfaces in terms of mapping class groups of punctured tori.
