  Let $F$ be a local non-Archimedean field of characteristic zero with a finite
residue field. Based on Tadi\'{c}'s classification of the unitary dual of
$\mathrm{GL}_{2n}(F)$, we classify irreducible unitary representations of
$\mathrm{GL}_{2n}(F)$ that have nonzero linear periods, in terms of Speh
representations that have nonzero periods. We also give a necessary and
sufficient condition for the existence of a nonzero linear period for a Speh
representation.
