Here's a quick demonstration of how the new experimental Mirror Translations button works. But I don't have video making capabilities, so you'll need to do the work yourself!
First, create a new flame and add a transform. Change the variation to dc_perlin and the amount from 1 to 5. Adjust the gradient and size so you have a textured box on the screen.
Add a Final transform and change the variation to loonie. (Mirror Translations works with normal transforms and other variations, but a final loonie makes a nice demonstration.) This will add a circle to the center of the box with the typical loonie distortion inside the circle.
Now, go to the Affine tab and make sure Post TF and Mirror Translations are not selected. Click the Move Triangles button (second from the top), make sure the loonie transform is selected, and move the triangle. The box moves around, but the loonie circle stays put in the center, so it affects different places in the box as it moves.
In the Affine tab, click Reset TF to re-center the box, and select Post TF. (Leave Mirror Translations unselected for now.) Move the triangle as in the previous step. The box again moves around, but the loonie circle moves along with it since the move is done after the variation.
In the Affine tab, click Reset TF to re-center the box again, and select Mirror Translations. (Leave Post TF selected as well; although translation mirroring works with both pre and post affine transforms, it works more logically with post transforms.) Now move the triangle. This time, the box stays put but the loonie circle moves around. This is the crux of the Mirror Translations functionality. Moving the triangle now changes both the pre and post affine transforms. The pre affine transform moves the box in relation to the loonie circle, so selects the part of the box the circle affects. Then the post affine transform moves the box back to its original position.
The effect depends a lot on the variations used, and works best with variations that are mostly linear, like loonie. Others to try are splits, waves2, glynnSim, collideoscope, and blob (increase low to something like 0.8 to see it better). It doesn't work as neatly for variations that center their output, like elliptic, eyefish, and julian. But don't let that stop you from trying!
It's considered experimental because the post doesn't exactly counter the pre if the post transform is rotated or scaled, or if the variation amount is not 1. But it can still be useful; just be aware that the behavior may change in future versions.