A student of mine recently came up with a programming solution that is an instance of $H^{-1} \circ T \circ H$, where $T$ is a translation and $H$ and $H^{-1}$ are roughly an embedding/projection pair. (The embedding and projection are not exact because in this example $T$'s domain and codomain are different.)
I'm pretty sure I've seen examples like this before, especially when $T$'s domain and codomain are equal. Maybe in quantum mechanics? But I have not been able to find any examples, and I have not been able to recall the name of this transformation, in which $H$ and its inverse are used to modify $T$.
Where can I find examples of this transformation? And perhaps learn what the transformation is called?
This is called a conjugation of $T$ by $H$. It’s a general abstract algebra concept that works in any group. An example from linear algebra is matrix similarity, which can function as a change of basis.