For a spherically symmetric Riemannian manifold, we can write the metric as
$ds^2 = a^2(r)dr^2 + b^2(r)d\Omega^2$
Now I think the metric components have to be even functions for the manifold to be smooth, because if you go through the origin to "negative $r$" you're really just going to positive $r$ on the other side, so you have to hit all the same metric values.
And that implies that $a(r)$ and $b(r)$ (what I'm calling the "metric functions" for lack of a precise term) would all have to be even or odd, right? I'm especially interested in the tangential function $b(r)$ -- to be smooth, we need $b(0) = 0$ and $b'(0) = 1$ so that it locally looks like Euclidean space, so I guess $b(r)$ has to be odd. Whereas the radial function, as well as the time function in a Lorentzian manifold, have to be even.
Is that correct?