Tetrahedron Permutations 
There are 2 identical regular tetrahedrons. Each is painted with 1 red face, 1 blue face, 1 green face, and 1 yellow face.
  Will they always be painted in the same way?
  Clarification: They are painted in the same way if you cannot tell them apart from each other.

This is the type of problem I can easily get through common sense / visualization but then feel I am cheating myself by not using any math. Is there a way to frame this as a permutation, perhaps by "unfolding" / flattening the tetrahedron and then viewing it as a rotation? 
I visualize it as a circular permutation of three items (three triangles) around an unchanging central item (since the rotation or reflection does not affect the central item). This would lead to (3-1)! Permuatations around the center / circle, i.e. 2 distinct arrangements. Technically any of them can be the center triangle depending on how we choose to flatten / unfold it. 
This is easy to conceptualize and visualize, but I'm not satisfied. The main reason is that in 2 dimensions it's possible to make 8 combinations (I think), because each of the 4 colors could be at the center, and each central color can have 2 distinct permutations in a circle around it. How can I mathematically account for the introduction of the 3rd dimension? Clearly eliminating one of the variables works but I'm not sure why yet...
Can someone help?
NOTE: I found this problem in the "Brilliant" app and got the correct answer, but feel there's something more to be learned here than I have taken from it. 
 A: You want to use math?  The symmetry group of a regular tetrahedron is isomorphic to $A_4.$
The ways to apply 4 paint colors to an asymmetric tetrahedron is isomorphic to $S_4$
$A_4$ is a normal subgroup of $S_4$  
There are 2 cosets.
A: If you place a tetrahedron on a surface so it has one of the sides at the bottom, then you can turn it to see one of the three other faces. Let's call such a turn a rotation.  And let's call it a flip when you place a different side at the bottom. So: a single tetrahedron allows for 3 rotations and 4 flips, resulting in 12 different 2D representations. Since there are 24 different colorings of the 2D representation, you indeed get 2 different paintings.
Now, the effect that a rotation has on a 2D representation is obvious: just rotate the colors on the outside. Indeed, this is why you didn't even consider all 24 different colorings, and just considered the 8 different ones that are invariant under rotation. 
On the other hand, the effect of a flip on a 2D representation is far less obvious. Indeed, given two 2D representations with a different color in the middle, can you quickly tell if they represent the same tetrahedron or not? Still, while you may have a hard time figuring out which are the same and which are different, you know for a fact that the same one tetrahedron can be represented with each one of the 4 sides in the middle. This is why the 8 needs to be divided by 4 to give 2 possible paintings, as you visually figured out as well.
