  A unique sink orientation (USO) is an orientation of the $n$-dimensional cube
graph ($n$-cube) such that every face (subcube) has a unique sink. The number
of unique sink orientations is $n^{\Theta(2^n)}$. If a cube orientation is not
a USO, it contains a pseudo unique sink orientation (PUSO): an orientation of
some subcube such that every proper face of it has a unique sink, but the
subcube itself hasn't. In this paper, we characterize and count PUSOs of the
$n$-cube. We show that PUSOs have a much more rigid structure than USOs and
that their number is between $2^{\Omega(2^{n-\log n})}$ and $2^{O(2^n)}$ which
is negligible compared to the number of USOs. As tools, we introduce and
characterize two new classes of USOs: border USOs (USOs that appear as facets
of PUSOs), and odd USOs which are dual to border USOs but easier to understand.
