  Considering uniform hypergraphs, we prove that for every non-negative integer
$h$ there exist two non-negative integers $k$ and $t$ with $k\leq t$ such that
two $h$-uniform hypergraphs ${\mathcal H}$ and ${\mathcal H}'$ on the same set
$V$ of vertices, with $| V| \geq t$, are equal up to complementation whenever
${\mathcal H}$ and ${\mathcal H}'$ are $k$-{hypomorphic up to complementation}.
Let $s(h)$ be the least integer $k$ such that the conclusion above holds and
let $v(h)$ be the least $t$ corresponding to $k=s(h)$. We prove that $s(h)=
h+2^{\lfloor \log_2 h\rfloor} $. In the special case $h=2^{\ell}$ or
$h=2^{\ell}+1$, we prove that $v(h)\leq s(h)+h$. The values $s(2)=4$ and
$v(2)=6$ were obtained in a previous work.
