We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative $L_p$-space for some $p>1$. This is a noncommutative version of Rosenthal's result for commutative $L_p$-spaces. Similarly for $1\le q<2$, an infinite-dimensional subspace $X$ of a noncommutative $L_q$-space either contains ${\ell }_q$ or embeds in $L_p$ for some $q<p<2$. The novelty in the noncommutative setting is a double-sided change of density