  Using a sample of $447.9\times 10^6$ $\psi(3686)$ events collected with the
BESIII detector in 2009 and 2012, we study the decays $\chi_{c0,2}\to$
$\eta'\eta'$ and $\eta\eta'$. The decays $\chi_{c2}\to\eta'\eta'$,
$\chi_{c0}\to\eta\eta'$ and $\chi_{c2}\to\eta\eta'$ are observed for the first
time with statistical significances of $9.6\sigma$, $13.4\sigma$ and
$7.5\sigma$, respectively. The branching fractions are determined to be
$\mathcal{B}(\chi_{c0}\to\eta'\eta') =(2.19\pm0.03\pm0.14)\times10^{-3}$,
$\mathcal{B}(\chi_{c2}\to\eta'\eta') = (4.76\pm0.56\pm0.38)\times10^{-5}$,
$\mathcal{B}(\chi_{c0}\to\eta\eta') = (8.92\pm0.84\pm0.65)\times10^{-5}$ and
$\mathcal{B}(\chi_{c2}\to\eta\eta') = (2.27\pm0.43\pm0.25)\times10^{-5}$, where
the first uncertainties are statistical and the second systematic. The
precision for the measurement of $\mathcal{B}(\chi_{c0}\to\eta'\eta')$ is
significantly improved compared to previous measurements. Based on the measured
branching fractions, the role played by the doubly and singly
Okubo-Zweig-Iizuka disconnected transition amplitudes for $\chi_{c0,2}$ decays
into pseudoscalar meson pairs can be clarified.
