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.
Expanding a bit of my comment into an answer, I would strongly suggest taking a different approach to your root problem: rather than geometrically determining the YPR values for each face, instead start by constructing the list of vertices for the pentagonal hexecontrahedron and represent your faces as lists of vertices. The vertices are easy to find: each corresponds to the center of a face on the snub dodecahedron, and since all the faces of that polyhedron are regular then the center of a face is just the mean of its vertices. Similarly, you can use the connectivity information you presumably have for the snub dodecahedron to build the same information for the hexecontrahedron: to determine the vertices that comprise a face, simply look at all the faces of the snub dodecahedron that include the hexecontrahedron's face's dual vertex.
Once you have the vertex and face information for the hexecontrahedron you can derive the orientational information (but for heaven's sake, use quaternions or rotation matrices instead of YPR unless you really have to! And if you do absolutely have to, you'll generally have more luck with the phrase 'Euler angles' than 'Tait-Bryan angles'), but for most purposes you're unlikely to actually need anything more than the vertex list and facial index list for the hexecontrahedron.
Best Answer
The $3$ Euler angles (usually denoted by $\alpha, \beta$ and $\gamma$) are often used to represent the current orientation of an aircraft.
Starting from the "parked on the ground with nose pointed North" orientation of the aircraft, we can apply rotations in the Z-X'-Z'' order:
to get the current orientation of the aircraft represented by the $3$ Euler angles $(\alpha, \beta, \gamma)$.
You may have noticed that we yaw twice, and we never use pitch. In fact, there are many ways of describing the orientation of an aircraft (or other rigid objection), some of which use all three: some amount $a$ of yaw, then some amount $b$ of pitch, then some amount $c$ of roll.
There exist standard formulas for converting between different ways of describing some given orientation: orientation in $(a, b, c)$ format; orientation in Euler angle $ ( \alpha, \beta, \gamma ) $ format; orientation as described by a $3 \times 3$ rotation matrix, etc.
I'm assuming one popular coordinate system in flight dynamics which associates: