Let $(W,S)$ be a finite Coxeter system, and let $J\,{\subseteq}\, S$. Any $w\,{\in}\,W$ has a unique factorization $w\,{=}\,w^Jw_J$, where $w_J$ belongs to the parabolic subgroup $W_J$ generated by $J$, and $w^J$ is of minimal length in the coset $wW_J$. It is shown here that $w_I\,{=}\,w^J$ if and only if $w^I\,{=}\,w_J$, for all $I,J\,{\subseteq}\,S$. Furthermore, a similar symmetry property in arbitrary $(W_I,W_J)$-double cosets is conjectured, which links this result to the Solomon descent algebra of $W$.