  An order relation for contractions on a Hilbert space can be introduced by
stating that $A\preccurlyeq B$ if and only $A$ is unitarily equivalent to the
restriction of $B$ to an invariant subspace. We discuss the equivalence classes
associated to this relation, and identify cases in which they coincide with
classes of unitary equivalence. The results extend those for completely
nonunitary partial isometries obtained by Garcia, Martin, and Ross.
