  We develop a notion of a `canonical $\mathcal{C}$-henselian valuation' for a
class $\mathcal{C}$ of field extensions, generalizing the construction of the
canonical henselian valuation of a field. We use this to show that the $p$-adic
valuation on a finite extension $F$ of $\mathbb{Q}_p$ can be recovered entirely
(or up to some indeterminacy of the residue field) from various small quotients
of $G_F$, the absolute Galois group of $F$. In particular, it can be recovered
fully from the maximal solvable quotient. We use this to prove several versions
of the birational section conjecture for varieties over $p$-adic fields.
