Every conversion I've seen for finding the axis and angle $\theta$ of a rotation from a rotation matrix $\mathbf{R}$ uses
$tr (\mathbf{R}) = 1+2\cos{\theta}$
and then inverting by way of $\cos^{-1}$. But $\cos{\theta}=\cos{(-\theta)}$. Therefore, if the only information we really have is $\cos\theta$, the rotation angle could be $\theta$, or just as easily, $-\theta$. The only ways I have figured to solve the ambiguity are rather clunky. Is there a short, straightforward way to do it?
You should take into account that matrix $R(v,\theta)=R(-v,-\theta)$.
So we have two possibilities $v$ and $-v$ for the axes and appropriately two possible values of the angle which have the same $cos(\theta) $ value.
You can calculate the axis from the formula: $v= {\dfrac {1}{2sin(\theta)}}\begin{bmatrix} r_{32}-r_{23} \\ r_{13}-r_{31} \\ r_{21} -r_{12} \end{bmatrix}$ where $r_{ij}$ are appropriate entries of $R$ matrix, so you see from this formula that changing sign of the angle $\theta$ changes sign of $sin(\theta)$ and consequently orientation of $v$ vector.