Extending harmonic function from half-space to whole space

Question

$u$ is harmonic in $\mathbb{R}^n_+$ and $u=0$ on the boundary. I wish to extend $u$ to a harmonic function on $\mathbb{R}^n$. Suppose I defined $u(x_1,...,x_{n-1},x_n)=u(x_1,...,x_{n-1},-x_n)$ for $x_n\leq0$. Clearly $u$ would be harmonic on the lower half space. But what about the boundary? If $u$ is harmonic at a point $\xi$ on the boundary it must satisfy the mean value property i.e. $$0=u(\xi)=\frac{1}{w_n}\int_{|x|=1}u(\xi+rx)dS_x$$ But for that to hold, shouldn't I define $u(x_1,...,x_{n-1},x_n)=-u(x_1,...,x_{n-1},-x_n)$?