  We investigate the structure of the twisted Brauer monoid $\mathcal
B_n^\tau$, comparing and contrasting it to the structure of the (untwisted)
Brauer monoid $\mathcal B_n$. We characterise Green's relations and pre-orders
on $\mathcal B_n^\tau$, describe the lattice of ideals, and give necessary and
sufficient conditions for an ideal to be idempotent-generated. We obtain
formulae for the rank (smallest size of a generating set) and (where
applicable) the idempotent rank (smallest size of an idempotent generating set)
of each principal ideal; in particular, when an ideal is idempotent-generated,
its rank and idempotent rank are equal. As an application of our results, we
also describe the idempotent-generated subsemigroup of $\mathcal B_n^\tau$
(which is not an ideal) as well as the singular ideal of $\mathcal B_n^\tau$
(which is neither principal nor idempotent-generated), and we deduce a result
of Maltcev and Mazorchuk that the singular part of the Brauer monoid $\mathcal
B_n$ is idempotent-generated.
