  Let $0<p<q\leq\infty$ and $\alpha \in (0,\infty]$. We give a characterization
of quasi-Banach interpolation spaces for the couple
$(L_p(0,\alpha),L_q(0,\alpha))$ in terms of two monotonicity properties,
extending known results which mainly dealt with Banach spaces. This enables us
to recover recent results of Cwikel and Nilsson on sequence spaces and to solve
a conjecture of Levitina, Sukochev and Zanin in the setting of function spaces.
We apply the results obtained to characterize symmetric spaces in which the
standard forms of the noncommutative Khintchine inequalities hold.
