When does the angle between the planes in tetrahedral molecule with central atom deviate from 90°?

Consider an atom X bonded to four other atoms A, B, C, D in a tetrahedral fashion with sp³ hybridisation. If A, B, C, D are the same, every bond angle is 109.47°, and the angle between the planes formed by joining nuclei of X, A, B and X, C, D is exactly 90° (treat nuclei as point sized since they are very small with respect to the atoms) if pairs A, B and C, D are identical elements (symmetrical molecules like methane or dichloromethane):


[![Ideal XABCD](https://i.stack.imgur.com/TnkJW.png)](https://i.stack.imgur.com/TnkJW.png)


If A, B, C, D are different, then the bond angle will be different to minimise repulsion and maximize stability as per VSEPR theory. But what happens to the angle between the planes? My intuition says that it will always be 90° in any molecule and the reason is that *repulsions are minimised in that case*.


However, this reasoning is only a qualitative one, but I wish that this could be mathematically proved or in some other way using some very good logic. In the following figure you can see that due to decrease in angle between bonding planes, B and D come close to each other and also A and C come close to each other. So they will face steric repulsion from each other leading to instability:


[![Skewed XABCD](https://i.stack.imgur.com/wGN7P.png)](https://i.stack.imgur.com/wGN7P.png)


In case the angle between the planes is not always 90°, what are these exceptions? In organic chemistry, many kind of exceptions frequently occur, so I was trying to think of some case where steric hindrance or torsional strain or angle strain or ring strain or some other reason may cause this angle to change.


The boat configuration of cyclohexane came into mind. I knew that there is flagpole interaction between two hydrogen atoms which get very close to each other. Maybe this repulsion may cause hydrogen atoms to deviate a little to the sides, causing *a little* change in angle between bonding planes in this example. This could be the first exception, but I am not very sure of it.


Also, such repulsion can happen in two ways: clockwise and counterclockwise rotation of the planes. For example, in the above figure a mirror image can be formed if B comes closer to C and D comes closer to A. Both of these compounds will be mirror image of each other and therefore may create optical activity. This optical activity if it happens can prove to be a practical confirmatory test to be sure weather this thinking is correct or not.


If I am correct, are there better examples where the angle between bonding planes is different from 90°?


