  The twisted partition monoid $\mathcal{P}_n^\Phi$ is an infinite monoid
obtained from the classical finite partition monoid $\mathcal{P}_n$ by taking
into account the number of floating components when multiplying partitions. The
main result of this paper is a complete description of the congruences on
$\mathcal{P}_n^\Phi$. The succinct encoding of a congruence, which we call a
C-pair, consists of a sequence of $n+1$ congruences on the additive monoid
$\mathbb{N}$ of natural numbers and a certain $(n+1)\times\mathbb{N}$ matrix.
We also give a description of the inclusion ordering of congruences in terms of
a lexicographic-like ordering on C-pairs. This is then used to classify
congruences on the finite $d$-twisted partition monoids
$\mathcal{P}_{n,d}^\Phi$, which are obtained by factoring out from
$\mathcal{P}_n^\Phi$ the ideal of all partitions with more than $d$ floating
components. Further applications of our results, elucidating the structure and
properties of the congruence lattices of the ($d$-)twisted partition monoids,
will be the subject of a future article.
