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Let's say we start with a vector $u = [0, 0, 1]$ which represents the positive z-axis.

Now, let's say we arbitrarily transform this vector via a rotation $z' = R \begin{bmatrix}0 \\ 0\\ 1 \end{bmatrix}$.

There are arbitrarily many rotations that can lead to the same $z'$.

I want to calculate a new rotation from $R$, $R_{undo}$, which inverts $R$ except it does not invert any portion of the rotation corresponding to a planar rotation around $z'$ (e.g. $z'$ is the normal vector of the plane of rotation).

For example, let's say in Euler angles with order zyx, $R = [90, 90, 0]$. Then, using the same zyx convention, $R_{undo} = [0, -90, 0]$, not $[-90, -90, 0]$.

How do we calculate $R_{undo}$ in general, for an arbitrary axis u?

user3180
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It seems to me that your $R_{\text{undo}}$ is not very well defined, and that we should assume everything is in $\mathbb{R}^3$. From your example, if $z'=Ru$ for some rotation $R$, it looks like you want $R_{\text{undo}}$ to be the unique rotation that maps $z'\mapsto u$ and fixes the plane spanned by the vectors $z'$ and $u$ (take a moment to think about what this means geometrically). But note that if $z'=-u$, it is not at all clear what $R_{\text{undo}}$ should do -- the vectors won't span a plane, and there is no "prefered direction" to bring $z'$ back to $u$ anymore.

So let's put this problem aside, and assume $z'\neq-u$. Then you can calculate $R_{\text{undo}}$ following the procedures in the answers to this question, which looks very much like yours:

Calculate Rotation Matrix to align Vector A to Vector B in 3d?