The distortions of coordination polyhedra have been extensively discussed in the literature. One of the relevant references has been published a while ago in Acta Cryst. (1998). B54, 766-773 (https://doi.org/10.1107/S0108768198003905).

As Andrzej says there are many ways this can be measured. A few ways are described in Chem. Soc. Rev., 2011, 40, 4119-4142. There's also a program called SHAPE which is available from the University of Barcelona which provides a very thorough estimate of the coordination polyhedra.

Manganite (y-MnOOH) is an important example of John-Teller distorted structure.

My question is too specific. angle between axial (O1 Mn2 O4 ) and equatorial (O2 Mn O8) are not 180deg. Then, what happens to degeneracy and splitting of orbitals.