  We say that a structure $\mathcal{A}$ admits \emph{strong jump inversion}
provided that for every oracle $X$, if $X'$ computes $D(\mathcal{C})'$ for some
$\mathcal{C}\cong\mathcal{A}$, then $X$ computes $D(\mathcal{B})$ for some
$\mathcal{B}\cong\mathcal{A}$. Jockusch and Soare \cite{JS} showed that there
are low linear orderings without computable copies, but Downey and Jockusch
\cite{DJ} showed that every Boolean algebra admits strong jump inversion. More
recently, D.\ Marker and R.\ Miller \cite{MM} have shown that all countable
models of $DCF_0$ (the theory of differentially closed fields of characteristic
$0$) admit strong jump inversion. We establish a general result with sufficient
conditions for a structure $\mathcal{A}$ to admit strong jump inversion. Our
conditions involve an enumeration of $B_1$-types, where these are made up of
formulas that are Boolean combinations of existential formulas. Our general
result applies to some familiar kinds of structures, including some classes of
linear orderings and trees. We do not get the result of Downey and Jockusch for
arbitrary Boolean algebras, but we do get a result for Boolean algebras with no
$1$-atom, with some extra information on the complexity of the isomorphism. Our
general result gives the result of Marker and Miller. In order to apply our
general result, we produce a computable enumeration of the types realized in
models of $DCF_0$. This also yields the fact that the saturated model of
$DCF_0$ has a decidable copy.
