Why can't the change in a crystal structure be due to the rotation of octahedra?

I have the following information about barium titanate ($\ce{BaTiO3}$).


It adopts the perovskite structure and
has a phase transition from a cubic to a tetragonal phase at 120 °C.
The space group of the cubic phase is Pm3m.
The space group of the tetragonal phase is P4mm.
Just above 120 °C the cubic cell parameter, a = 4.01 Å;
at room temperature, the cell parameters of the tetragonal phase are a = 3.992 Å and c = 4.034 Å.


Apparently the phase transition occurs because of a displacement of the $\ce{Ti}$ atom; why can it not be due to rotations of the $\ce{TiO6}$ octahedra?




Below is a depiction of the two unit cells ([source](https://www.researchgate.net/publication/334759421_Preparation_and_Physical_Properties_of_Barium_Titanate_with_Some_Added_Oxides/figures?lo=1))


[![enter image description here](https://i.stack.imgur.com/8BjLW.jpg)](https://i.stack.imgur.com/8BjLW.jpg)


Apparently, the entire structure stretches in one direction (and scrunches in the other two), with the c-axis 2.5% longer than the other two in the tetragonal structure ([source](https://materialsproject.org/materials/mp-5986)). As a consequence, the octahedral oxygen configuration is now a distorted octahedron. In addition, the titanium is no longer centered in the octahedron, leading to a dipole moment along the c-axis ([source](https://www.doitpoms.ac.uk/tlplib/ferroelectrics/phase_changes.php)).


However, all these movements are translations along the c-axis. They preserve the symmetry of the tetragonal space group. If you were to rotate the (distorted) oxygen octahedra, the tetragonal symmetry (which includes mirror planes perpendicular to a and to the a+b diagonal) would break down.


In comparing different views of the unit cell, be aware that some authors place the titanium on the origin (with barium in the center), while others choose to place the barium on the origin (with titanium in the center).



> 
> Why can't the change in a crystal structure be due to the rotation of octahedra?
> 
> 
> 


It could, but not for the transition to the tetrahedral crystal form. There are less symmetric crystal forms under other conditions, and these would not have the symmetry constraints.


UPDATE: The cubic structure seems to have some added complexity, with 8 possible sites for a disordered titanium atom in the cubic structure.


[![enter image description here](https://i.stack.imgur.com/iE1T6m.png)](https://i.stack.imgur.com/iE1T6m.png)


Source: [doi:10.1103/PhysRevB.70.134107](https://doi.org/10.1103/PhysRevB.70.134107)


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