We introduce a sorting machine consisting of $k+1$ stacks in series: the first $k$ stacks can only contain elements in decreasing order from top to bottom, while the last one has the opposite restriction. This device generalizes the $\mathfrak{DI}$ machine introduced by Rebecca Smith, which studies the case $k=1$. Here we show that, for $k=2$, the set of sortable permutations is a class with infinite basis, by explicitly finding an antichain of minimal nonsortable permutations. This construction can easily be adapted to each $k\geqslant 3$. Next we describe an optimal sorting algorithm, again for the case $k=2$. We then analyze two types of left-greedy sorting procedures, obtaining complete results in one case and only some partial results in the other one. We close the paper by discussing a few open questions.