  Using an off-critical deformation of the identity of Duminil-Copin and
Smirnov, we prove a relationship between half-plane surface critical exponents
$\gamma_1$ and $\gamma_{11}$ as well as wedge critical exponents
$\gamma_2(\alpha)$ and $\gamma_{21}(\alpha)$ and the exponent characterising
the winding angle distribution of the O($n$) model in the half-plane, or more
generally in a wedge of wedge-angle $\alpha.$ We assume only the existence of
these exponents and, for some values of $n,$ the conjectured value of the
critical point. If we assume their values as predicted by conformal field
theory, one gets complete agreement with the conjectured winding angle
distribution, as obtained by CFT and Coulomb gas arguments. We also prove the
exponent inequality $\gamma_1-\gamma_{11} \ge 1,$ and its extension
$\gamma_2(\alpha)-\gamma_{21}(\alpha) \ge 1$ for the edge exponents. We provide
conjectured values for all exponents for $n \in [-2,2).$
