We are given 3 real eigenvalues $\lambda_1 > \lambda_2 > \lambda_3$ of a $3 \times 3$ real symmetric matrix $A$.
Which are the extra real parameters that are needed in order to reconstruct $A$ or any rotated version $R(\theta)AR(\theta)^T$ of it, and what's the process to reconstruct it?
A real symmetric matrix with distinct eigenvalues has orthogonal eigenvectors, so determining $A$ comes down to choosing an orthonormal basis for $\mathbb{R}^3$.
In other words choose a $U \in O(3)$ and set $A = UDU^T$ where $D$ is the matrix with $\lambda_i$ along the diagonal.
If by rotation you mean an element of $SO(3)$ then there are only two possible choices since $O(3)/SO(3) = \{\pm I\}$. The choices corresponds to the sign of the permutation of the eigenvalues.
If you mean rotation around a specific axis then it is a little more complicated. Consider the angle the eigenvector corresponding to $\lambda_1$ makes with the axis and call it $\alpha \in [0, \pi)$. If $\alpha=0$ then the only necessary choice is the sign of the permutation of $\lambda_i$. If $\alpha \neq 0$ then the eigenvector is uniquely determined by $\alpha$ and the rotation angle, so any choice of two other eigenvectors orthogonal to the first should give a unique matrix. So there are the same amount of possibilities as matrices in $O(2)$. $O(2)$ is parametrized by an angle in $[0, 2\pi)$ and a sign.
So the parametrization looks like $(\alpha, \theta, s) \in \Big([0, \pi) \times [0, 2\pi)/((0, x) \sim (0, y))\Big) \times \{\pm 1\}$.