$ E(f(X_0)f'(X_l)) = E(f'(X_0) f(X_l)) $ for a stationary process?

I have a stationary process ${X_t}_{tinmathbb N}$ on $mathbb R$ and two functions $f, f’: mathbb R rightarrow mathbb R$.

Does the equation
$$ E(f(X_0)f'(X_l)) = E(f'(X_0) f(X_l)) $$
hold?

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Do I need strict stationarity or does weak stationarity suffice for the equation to hold?

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I tried to show
$$
P(f(X_0)f'(X_l) > x) = P(f'(X_0)f(X_l) > x)
$$
but didn’t succeed so far.