  We give a positive answer to a question raised by Davis et al. ({\em Discrete
Mathematics} 341, 2018), concerning permutations with the same pinnacle set.
Given $\pi\in S_n$, a {\em pinnacle} of $\pi$ is an element $\pi_i$ ($i\neq
1,n$) such that $\pi_{i-1}<\pi_i>\pi_{i+1}$. The question is: given
$\pi,\pi'\in S_n$ with the same pinnacle set $S$, is there a sequence of
operations that transforms $\pi$ into $\pi'$ such that all the intermediate
permutations have pinnacle set $S$? We introduce {\em balanced reversals},
defined as reversals that do not modify the pinnacle set of the permutation to
which they are applied. Then we show that $\pi$ may be sorted by balanced
reversals (i.e. transformed into a standard permutation $\Id_S$), implying that
$\pi$ may be transformed into $\pi'$ using at most $4n-2\min\{p,3\}$ balanced
reversals, where $p=|S|\geq 1$. In case $p=0$, at most $2n-1$ balanced
reversals are needed.
