Any proper rotation (in three dimensions) can be expressed using the Tait-Bryan (sometimes called improper Euler) angles in the form $$ R(\phi,\theta,\psi)=R_z(\phi)R_y(\theta)R_x(\psi) $$ where $R_z(\phi)$ is the a rotation by $\phi$ about the Z axis, and similarly for the other terms.
Let's define a conjugate rotation $R^C$ by the rotation expressed using the negatives of those same Euler angles: $$ [R(\phi,\theta,\psi)]^C=R_z(-\phi)R_y(-\theta)R_x(-\psi) $$ Now let's define an "undone rotation" as the result of somebody trying to undo a rotation $R$ by multiplying $R^C$. That is, $$ U(\phi,\theta,\psi) = R_z(\phi)R_y(\theta)R_x(\psi)R_z(-\phi)R_y(-\theta)R_x(-\psi) $$ The set $\left\{U\right\}$ of all undone rotations (defined in this way) is labeled by three real parameters, and all members are proper rotations, but it is by no means clear that every rotation can be expressed as a member of $\left\{U\right\}$. In fact, but I have not been able to find a set of $(\phi,\theta,\psi)$ such that $U(\phi,\theta,\psi) = R_x(\alpha)$ for arbitrary values of $\alpha$.
My question is, does the set $\left\{U\right\}$ cover the entire rotation group? If not, is there an easy way to characterize those rotations which are or are not a member of $\left\{U\right\}$? And if $\left\{U\right\}$ is not the whole rotation group, is it a subgroup of it?