  Let $L_{K}(A)$ be the free Lie algebra on a finite alphabet $A$ over a
commutative ring $K$ with unity. For a word $u$ in the free monoid $A^{*}$ let
$\tilde{u}$ denote its reversal. Two words in $A^{*}$ are called twin (resp.
anti-twin) if they appear with equal (resp. opposite) coefficients in each Lie
polynomial. Let $l$ denote the left-normed Lie bracketing and $\lambda$ be its
adjoint map with respect to the canonical scalar product on the corresponding
free associative algebra. Studying the kernel of $\lambda$ and using several
techniques from combinatorics on words and the shuffle algebra, we show that
when $K$ is of characteristic zero two words $u$ and $v$ of common length $n$
that lie in the support of ${\mathcal L}_{K}(A)$ - i.e., they are neither
powers $a^{n}$ of letters $a \in A$ with exponent $n > 1$ nor palindromes of
even length - are twin (resp. anti-twin) if and only if $u = v$ or $u =
\tilde{v}$ and $n$ is odd (resp. $u = \tilde{v}$ and $n$ is even).
