What is the purpose of Rust's one-eye mirror?

The Eye is the window to the soul...

The mirror allows Rust to 'see' his own soul, to meditate and focus on his own consciousness and gain prescience, even enlightenment... internally, of course. Cohle is a vehement atheist, but he still believes in a higher level of consciousness: whether borne of delusion, drug induced hallucination or otherwise.

Previously during the series, Rust remarks about how, when looking into the eyes of a dead person, you can see that in the end "they welcome death'. He also has a fixation on suicide, yet 'lacks the constitution' to carry it out. He is looking into his eyes to 'sanity check' himself, to see if he is suicidal.

A mirror which only reflects an eye can only reflect one's soul, not the distracting vessel it occupies. It is a focal point for him...

I actually did a blog post about this where I used the theories of Roger Sperry and Carl Jung to conclude that the soul resides in the left eye and Rust thus uses his left eye mirror to reflect upon his own soul as a form of meditation:

...We all have two brain hemispheres. Left and right, which are connected by the corpus callosum, a bridge that connects the two hemispheres. Our right hemisphere controls the left side of our body and the left hemisphere controls the right side of our body.

Roger Sperry conducted ‘split brain’ experiments by removing the corpus callosum. After removing the link between the two hemispheres, Sperry proceeded to test the two hemispheres independently. [...] Sperry did more studies that further proved to him that thoughts are localised in different sides of the brain. Either right or left. [...] What this showed Sperry was that there are two modes of thought, Verbal and Non Verbal. I interpret verbal and non verbal to Instinctual and Intellectual...

...Contemporary society at large is left brained (yes, a mass generalisation) so most constructed identities you see in the urban capitalistic city environment are left brained Identities/Personalities. Imagine the opposite of a money driven capitalistic society, Bingo! an anthropological primitive but predominantly right brained society (generalised to prove a point again); [...] The right brain is minimally used in most social interactions due to the identity we project to others. The left brains allow people to be fake, calculated, ‘civilized’ e.g pretending to be nice when actually not. [...]

“Interestingly, not only are we not aware of our right brain identities, we can’t even control them! The right brain is non verbal and in touch with the unconscious, which is also non verbal. Many prominent psychologist such as Jung and Dr. Scott Peck, explain that the unconscious could be another term for the soul, or at least the window to it. Because of the crossover, our LEFT eye goes to our right brain. The left eye therefore, is literally the window to the soul and the indicator of the hidden, unmasked true self.

http://www.scienceofsoulmates.com/The_power_of_the_left_eye.htm

The right brain is non verbal and in touch with the unconsciousness or even the collective unconsciousness. While reading one of my favourite Psychiatrists Carl Jung, he convinced me that the word “Soul” can substitute the more scientific sounding, unconscious.

Therefore, when Rust says to Marty [in True Detective] “I know who I am”, what the average consumer didn’t realise was (like Marty), Rust meant he knew himself on multiple dimensions. Not just a vague ‘I know myself fooh!’ Nah, Rust truly knows himself inside out, more importantly Rust Cohle knows his Soul! This can be concluded by besides the many times he uses the word soul throughout the series...

• The only mirror Rust has is a coin sized mirror. I propose that the mirror is used as a form of meditation just as he uses the crucifix in his apartment as a form of meditation.

...Basically, your soul is in your left eye… Rust out of all knows this… Man.

I agree with @JohnSmithOptional's answer.

Only source I could site (embarrassingly) is Neil Strauss' The Game where the author is taught by Steve P. a PUA guru specializing in Hypnotism. They practice something they call Anamchara, staring into each others right eye, and this soul gazing creates a deep bond. Anamchara being the Gaelic meaning for friend of the Soul, or more literally Soul Friend.