Consider a polar angle $\theta$ and an azimuthal angle $\phi$ describing a vector in spherical coordinates, where $\theta \in \left(\pi, 2\pi\right]$ and $\phi \in \left[0, 2\pi\right]$.
Does there exist a convenient way to convert this pair of angles to an "equivalent" pair of angles where $\theta \in \left[0, \pi\right]$ and $\phi \in \left[0, 2\pi\right]$? "Equivalent" here means that the pair of angles represents the same vector, or in other words that converting from spherical to Cartesian coordinates will yield the same result. For instance, $\theta=\frac{5\pi}{4}, \phi=\frac{\pi}{4}$ is equivalent to $\theta=\frac{3\pi}{4}, \phi=\frac{5\pi}{4}$ because converting a vector of length 1 defined by either pair of angles yields the same $x,$ $y,$ and $z$ values, and $0 \leq \theta \leq \pi$.
If $\theta\in(\pi,2\pi]$ and $r=1$:
$$z=\cos\theta$$ so pick $\theta' = 2\pi-\theta$ to maintain the $z$ value.
$$x = \sin\theta \cos\phi = -\sin\theta' \cos\phi\\ y = \sin\theta \sin\phi = -\sin\theta' \sin\phi $$
so pick one of $\phi' = \phi \pm \pi$ to maintain the $x,y$ values.
$r, \theta,\phi,x,y,z$ are based on the convention given in Wikipedia: