  A \emph{bipartite monoid} is a commutative monoid $\Q$ together with an
identified subset $\P \subset \Q$. In this paper we study a class of bipartite
monoids, known as \emph{mis\`ere quotients}, that are naturally associated to
impartial combinatorial games.
  We introduce a structure theory for mis\`ere quotients with $|\P| = 2$, and
give a complete classification of all such quotients up to isomorphism. One
consequence is that if $|\P| = 2$ and $\Q$ is finite, then $|\Q| = 2^n+2$ or
$2^n+4$.
  We then develop computational techniques for enumerating mis\`ere quotients
of small order, and apply them to count the number of non-isomorphic quotients
of order at most~18. We also include a manual proof that there is exactly one
quotient of order~8.
