As the maker of the True Mirror, and with over 30,000 individual showings over the last 30 years, there's a big reason to get over the strange hand/eye dis-coordination that occurs (and also the unfortunate doubling effect of any asymmetries): It's really "us" looking back, our face works this way, and stops working when reversed. We can see how we are, not just what we look like.
The person that is looking back with a reversed set of eyes is not able to communicate naturally. This is because the messages within our face that augment our words (and/or our non-verbal messages as well) are easily read by anyone looking at us, forming a key component of communication. Tke key here is the messages are different left to right, just as the brain hemispheres are different left to right.
When you flip your face in a mirror, indeed the left and right sides are swapped...the argument that mirrors only flip front and back and the mirror left side is still on "your" left side is not correct or relevant. The fact that you are facing yourself means that the right side needs to be on the opposite of your right side and vice versa for the left. (you can't shake your own hand in a mirror, the left hand of the mirror person comes out to meet your right hand)
In a True Mirror, you are seeing your face with the eyes and features correctly positioned, just like anyone looking at you, and therefore the messages read correctly and your expressions communicate accurately. Your subsequent responses can then be normal, just as another person communicates with you. In any mirror, you are in an incredibly dynamic communication feedback loop with yourself. But in a reverse mirror, there is information distortion introduced into the message, which causes them to be read differently, and in practice causes expressions to fade very quickly. It manifests as more of a stare, especially in the eyes. In the True Mirror, the feedback loop is still there, but without the information distortion, and expressions stick around and even grow.
The key expression that is impossible to keep for more than a second or two, and which works marvelously in the True Mirror, is our genuine smile. Its the light in the eyes that is so different when you reverse them - they just don't carry the reason for the smile, causing them to fade rapidly. Whereas in the True Mirror, it actually can grow, just as they do in real life when we are talking with others. Smiles are amazing vehicles for communicating so many elements of empathy, humor, understanding, being ok and beauty.
We all have been living with a literal doppelganger in our own personal experience, a version of us that only we see, and which is missing so many parts of our expressive personality. We are used to it, but at some level, it is causing strange and unexamined effects on our self-image, understanding, awareness, esteem, confidence, and other self-ism, and possibly is a root cause of feeling different than others. With a feedback loop with distortion, you don't know where its going to go, so everyone seems to have a different relationship with themselves, usually at odds with how others relate to them.
So yes, in a True Mirror, the physical effects are challenging, but easily overcome with familiarization. I can shave in it now, and find a spec of stuff on my shirt without thinking, and likewise if I put on makeup, it would also become normalized (though I might wait a while before working on my eyes!)
But so much more important is the idea of getting familiar with and learning about the person that I am in real life, the person that everyone else sees, the person that has many expressions. And through the exquisite communication tool that is my face, this true reflection can clue me into who and how I am doing. It also suggests to stop looking in mirrors and having an altered being with a non-expressive face inform me of the same.
For the last two years I have been posting reactions on social media (@truemirroco), and so you can see for yourself how much more dynamic people can be with themselves in a non-reversing mirror, and how much gets lost within seconds when they meet their eyes in reverse.
Let me know if this makes sense and you have comments or questions!