The $K^+$ and $K^0$ meson mass difference induces the mixing of the $a^0_0(980)$ and $f_0(980)$ resonances, the amplitude of which, between the $K^+K^-$ and $K^0\bar K^0$ thresholds, is large in magnitude, of the order of $ m_{K}\sqrt{m^2_{K^0}-m^2_{K^+}}\approx\sqrt\alpha m^2_K$, and possesses the phase sharply varying by about 90$^\circ$. We suggest performing the polarized target experiments on the reaction $\pi^-p\to\eta\pi^0n$ at high energy in which the fact of the existence of

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... -f_0(980)$ mixing can be unambiguously and very easily established through the presence of a strong jump in the azimuthal asymmetry of the $\eta\pi^0$ $S$ wave production cross section near the $K\bar K$ thresholds. The presented estimates of the polarization effect to be expected in experiment are to a great extent model independent.