I hesitate to say it's completely
understood, but it's definitely well understood and much of it is encapsulated in the Fresnel equations
. I once derived these in an electrodynamics course, which was a neat experience although quite time consuming.
Anyway yes, it is possible to do interesting and weird things like that. For example you can make a lens out of some plastic that has the exact same refractive index as water (n=1.33), and if you place it in water it will become invisible because the plastic and water are no different as far as the light is concerned. This is also related to why your vision gets blurry under water. The difference in refractive index between your eye and water is less than between your eye and air, so when underwater your eye is unable to focus the light enough to form an image on the retina.
Turning a mirror transparent by changing the surrounding medium is a bit more complicated. Reflection from a typical mirror is actually not because of the glass, but by a metal film (the glass just protects it), and the high reflectivity happens because of how light interacts with free electrons within the metal. The incoming wave gets absorbed very quickly by driving the electrons to oscillate, and they also re-emit the wave backwards. (This also means light does penetrate metal to a very shallow but nonzero depth, which is why sufficiently thin metal films are transparent, like the gold coated face plates the astronauts use). Anyway I don't think it's possible to turn this sort of mirror transparent by changing the medium since the wave will still interact with the metal in the same way.
However, we can consider the reflection from something like glass instead. It is possible to completely negate this reflection by coating it with a carefully designed layer of the correct thickness and refractive index. This is more weird the more you think about it. The glass by itself both transmits and reflects light, but by adding something in front you can make it reflect less, and therefore transmit more? This actually has useful applications with coatings that improve the performance of camera lenses.
Here's another fun thing. Have you ever held a prism and noticed that at certain angles you cannot see anything behind it, and it's like the inside surface acts like a perfect mirror? This is total internal reflection. That in itself is interesting and the basis of fiber optics, but not too weird or unknown. What's weird is that if you press your thumb up against the back surface where it is acting like a mirror, then you can see just your thumb print against it. (Try this if you've never seen it). This effect is called "frustrated
total internal reflection".
Why is it weird? The light was being perfectly reflected inside the prism, except now you have your thumb against it with a very thin gap between the two. The size of the gap is crucial. If it is thin enough the light wave hits the inside edge of the prism and then tunnels
across this gap and into your thumb. Exactly like the quantum tunneling of electrons across a barrier (in fact it is the same physics)! Because the gap is thin enough for the light to tunnel across, it no longer reflects, and this is why you can see your thumbprint against what would otherwise act as a perfect mirror.