  In the Einstein--Podolsky--Rosen experiment, when Alice makes a measurement
on her part of a bipartite system, Bob's part is collapsed to, or steered to, a
specific ensemble. Moreover, by reading her measurement outcome, Alice can
specify which state in the ensemble Bob's system is steered to and with which
probability. The possible states that Alice can steer Bob's system to are
called steered states. In this work, we study the subset of steered states
which are pure after normalisation. We illustrate that these pure steered
states, if they exist, often carry interesting information about the shared
bipartite state. This information content becomes particularly clear when we
study the purification of the shared state. Some applications are discussed.
These include a generalisation of the fundamental lemma in the so-called
`all-versus-nothing proof of steerability' for systems of arbitrary dimension.
