Assume we have a smooth Riemannian metric $g$ on a small one-sided neighborhood $U$ of $0$ on the plane, say $U\_\epsilon=\lbrace(x, y): x^2+y^2<\epsilon, y\geq 0\rbrace$.


When does there exist a diffeomorphism $f$ of ( may be smaller) one-sided neighborhood of zero $U\_\delta$ such that the resulting Riemannian metric $f\_\star g$ can be smoothly extended $\textbf{by relflection}$ along the axis $y=0$.


In particular( but not only) I am mostly interested in cases:


$a)$ $g=\rho(x,y) g\_0$, where $\rho(x,y)$ is a conformal factor and $g\_0$ is standard Euclidean metric.


$b)$ Case $a$ with flat $g$, i.e. $\Delta\ln\rho=0$



Local boundary symmetrisation of Riemannian metrics by coordinate changes
In other words, you are asking the following:



> 
> Assume we cut a surface by a curve $\gamma$ and take doubling of one piece. 
>  When the doubling has smooth metric?
> 
> 
> 


Once you reformulate it this way, the answer is evident.
First note that $\gamma$ has to be geodesic otherwise the doubling has infinite curvature along the gluing. 


Further, you may write the metric near $\gamma$ in semigeodesic coordinates;
one coordinate, say $u\ge 0$, is the distance to the geodesic and the other, say $v$, is the parameter of its foot point. 
In this coordinates the metric has form $(\begin{smallmatrix}1&0\\\ 0&a\end{smallmatrix})$,
where $a=a(u,v)$ is a positive function.


The metric on the doubling is $C^\infty$-smooth iff the function 
$b(u,v)=a(|u|,v)$ 
is $C^\infty$-smooth near the $v$-axis. 


