Matrix rows or columns are traditionally listed under $(x,y,z)$ order.
Cyclically change the pairs under consideration i.e $(x,y)\to(y,z)\to(z,x)$. The pairs $(x,y)$ and $(y,z)$ show up in the same order in the matrix but the $(z,x)$ shows up in reverse in the matrix. That is the cause of apparent discrepancy but really there is no discrepancy.
For example write
$x'=x\cos \alpha - y \sin \alpha$
$y'=x\sin \alpha + y \cos \alpha$
now change $(x,y)\to(y,z)\to(z,x)$ and $\alpha\to \beta \to \gamma$ and write the three matrices to see how $(z,x)$ part gets flipped.
Edit:
If you want them to look alike then give up the matrix notation and instead write
$y'=y\cos \beta - z \sin \beta$
$z'=y\sin \beta + z \cos \beta$
And
$z'=z\cos \gamma - x \sin \gamma$
$x'=z\sin \gamma + x \cos \gamma$
In each instance if you try to write $\left[ \matrix{ x' \cr y' \cr z'}\right]$ in terms of $\left[ \matrix{ x \cr y \cr z}\right]$ you will see that the mystery goes away.
If $R_x$ rotates around the $x$-axis, and $R_y$ rotates around the $y$-axis, and you want to rotate first around $x$, and then around $y$, simply apply $R_y R_x$ to your vector, let's call it $v$.
This is because $R_x v$ rotates $v$ around the $x$-axis, then $R_y(R_x v)$ rotates $R_x v$ around the $y$-axis.
Best Answer
Is your stiffness matrix having some formula like the $ D $ operator defined in https://en.wikipedia.org/wiki/Stiffness_matrix#Practical_assembly_of_the_stiffness_matrix? In that case, it suggests a formula: each column will rotate like a vector. So you will have $ K' = A K $.
Regrettably, I am missing some context and particularly I don't know what a stiffness matrix is. But your answer will be found by understanding what the domain and range of $ K $ are; or in other words, understanding the physical meaning of the matrix. Suppose $ K: V \rightarrow W $, and a rotation $ R $ acts on $ V $ by matrix $ R_v $ and on $ W $ by a matrix $ R_w $, then the transformation for $ K $ is $ R_w K R_v^{-1} $.