After appending my 2019-04-30 update to my much older answer, the above excellent, to-the-point and from-the-trenches expert answer came in. I immediately changed the designated answer from mine to the new one. There's probably still some fun reading below in my old reply and update, though... :)
Alas, I must answer my own question: I found a very explicit example online description of someone who created a thick-film transmission hologram of a convex mirror. She (or he) describes seeing her own face clearly, even if only in monochrome. So, if I accept this description at face value, it clearly is possible to create a realistic mirror using only wave-exclusion diffraction effects. Cool!
Also, I am amused (or is it chagrined?) that this reminded me of the importance of reading long articles all the way to the end, even if you feel you already got the point. This description of an actual holographic mirror was hidden at the very end of the long posting on I mentioned in my question about how transmission holograms cannot form mirrors.
2019-04-30 Update
As noted in the comments below, the above link to an explicit description of a holographic mirror unfortunately is no longer available, not even in Internet archives.
However, this draft book chapter PDF on reflection using Denisyuk transmission holograms seems to provide pretty good coverage of the issues.
Still, as I get older I find I like finding the simplest possible explanations of things. The simplest proof that true holograhic mirrors can exist is this: You can see your own face in a pool of calm water.
Why? Well, the reason why thick film holograms can reflect light at all is because any change in refractive index in a transparent medium creates an amplitude -- a probability -- for light to be reflected back in the direction in which it came. Metal mirrors are just extreme examples of this effect, since the Fermi surface electrons in metals create a nearly 100% probability that photons will be reflected.
The quantum mechanical details of reflections works in transparent materials are covered delightfully in my favorite Richard Feynman book, QED: The Strange Theory of Light and Matter. In addition to its relevance here for understanding what is possible with holograms, I recommend QED strongly to anyone interested in understanding just how utterly and completely weird quantum mechanics really is.
Feynman discusses how properly space layers of changes in refractive index can create a surface that, at least for certain frequencies, has a nearly 100% probability of reflecting light. A holographic mirror!
Finally, take a contemplative look at this image (or a real example from your kitchen) of a roll of very layers of Mylar film:
Nearly everyone has at sometime noticed at some level of consciousness how remarkably metallic such rolls look, almost like aluminum foil. That is because even though the distances between the film layers are not wave-coherent as they would in a photographic hologram, they do collectively reflect more and more light, until the surface looks remarkably metallic... which is to say, remarkably like a mirror.
Such a roll of Mylar film thus can plausibly be construed as a crude mechanically constructed hologram, and thus a proof that at least at some level of quality, transparent materials can indeed be configured to create plausibly effective, metallic-looking reflective mirrors.
Best Answer
The Wikipedia article that Anna mentioned is an excellent description of holography and I'm not going to try and compete with it, but since I'm guessing that you're not a physicist the following might help make things clearer.
When you "see" things, you see them because they enter your eye, get focussed by the lens and hit the retina. In addition you see 3D because the image recorded by your left and right eyes are slightly different, and the brain can reconstruct a 3D image from the differences.
So if you're looking at a mouse (to use the example from the Wikipedia article) it's the light reflected off the mouse that the eye uses to "see" the mouse. Suppose you could come up with some clever trick to remove the mouse but still send the light to your eyes as if it had come from the mouse. Your brain couldn't tell the difference because your eyes are still receiving the same light as when the mouse was there. This is what a hologram does.
A hologram is a pattern of light and dark areas. When you shine a laser onto a hologram the light and dark areas scatter the light by a process called diffraction. The clever bit is that the light is scattered in exactly the same way as if there were a mouse there, so your brain sees light that looks as if it has come from a mouse, so you see a mouse. It appears in 3D because the hologram scatters light differently depending on the angle you're looking at it, so your left and right eye receive differently scattered light just as they would from a real mouse.
You might think it would be tremndously difficult to make a hologram to scatter light in just the right way to make it appear as a mouse, but actually you make a hologram from a real mouse i.e. it's just a type of photograph.
It's hard to make multicoloured holograms because to "see" the hologram you have to shine a laser on it, and lasers are just a single colour. You could use three lasers, e.g. a red, green and blue laser, but annoyingly the hologram scatters different coloured light in different ways and your multicoloured hologram would be very blurred.
I hope this helps - to get any further you'll need to work through the Wikipedia article, and also understand what diffraction is.