  The Galois action on the pro-$\ell$ \'etale fundamental groupoid of the
projective line minus three points with rational base points gives rise to a
non-commutative formal power series in two variables with $\ell$-adic
coefficients, called the $\ell$-adic Galois associator. In the present paper,
we focus on how Landen's functional equation of trilogarithms and its
$\ell$-adic Galois analog can be derived algebraically from the $S_3$-symmetry
of the projective line minus three points. Twofold proofs of the functional
equation will be presented, one is based on Zagier's tensor criterion devised
in the framework of graded Lie algebras and the other is based on the chain
rule for the associator power series. In the course of the second proof, we are
led to investigate $\ell$-adic Galois multiple polylogarithms appearing as
regular coefficients of the $\ell$-adic Galois associator. As an application,
we show an $\ell$-adic Galois analog of Oi-Ueno's functional equation between
$Li_{1,\dots,1,2}(1-z)$ and $Li_k(z)$'s $(k=1,2,...)$ .
