  It is sometimes desirable in choiceless constructions of set theory that one
iteratively extends some ground model without adding new sets of ordinals after
the first extension. Pushing this further, one may wish to have models $V
\subseteq M \subseteq N$ of $\mathsf{ZF}$ such that $N$ contains no subsets of
$V$ that do not already appear in $M$. We isolate, in the case that $M$ and $N$
are symmetric extensions (particular inner models of a generic extension of
$V$), the exact conditions that cause this behaviour and show how it can
broadly be applied to many known constructions. We call this behaviour upwards
homogeneity.
