Today I was minding my own business and for no particular reason suddenly turned my head to the left to see someone looking right at me. Not the first time something like that has happened.
Did my brain know someone was looking at me even though it didn’t consciously register? I would say the person was about 60-75° off axis from straight ahead. You hear a lot of people say “I just felt someone looking at me” and I wonder if this is the rational cognitive explanation.
In a “normal” human, the horizontal field of vision extends 200 degrees or more.
https://www.aao.org/education/image/normal-vision-field
So without consciously looking, you can detect subtle cues that someone/something is looking at you. Evolutionarily useful.
I’ve had it happen often in highway driving. Glance at someone in a car in the passing lane when they’re looking straight ahead, and they instantly turn to look at you.
I knew that our total field of vision was about that wide, but we don’t see color well there, acuity is not very good, and not much registers unless it’s moving. I don’t think we can read anything more than 45° off. It’s mostly a question about what the subconscious brain is doing while we’re busy watching cat videos.
From Wiki:
" Blindsight is the ability of people who are cortically blind to respond to visual stimuli that they do not consciously see due to lesions in the primary visual cortex,"
The upshot is that there seems to be some visual processing taking place outside your conscious sight.
Most of you never threw, or received, paper airplanes in a university lecture. In the audience, paper airplanes are a harmless and silent event that adds beauty without interrupting. For a lecturer, checking over his shoulder, paper airplanes are an incoming missile on the periphery of vision, B&W, low resolution.
Indeed the peripheral vision is especially sensitive to movement, having a greater density of rod cells than the center.
So the kind of phenomenon in the OP is common, because you notice movement but don’t know what it is until you turn your head and/or eyes towards the target.
To read text, the image has to fall on your fovea, so much closer to your center of vision than 45 degrees. You really need to be looking directly at it, within a circle not more than about 5 degrees in diameter.
I am of the opinion that humans are “pattern monkeys”, we see things and try to create patterns that match them. So we see eyes in our peripheral vision and realize they are focussed on us. (an important survival trait). Even in old paintings, the eyes “follow you around the room” because that is the pattern we see. it’s also why we think we see faces in the drk, or Jesus on our burnt toast, because our mind instinctively wants to “connect the dots” to match something familiar.
I used to have a monitor at my work which was far to the right of me on another bench. I never noticed it while working at the monitor in front of me or a piece of equipment. But when the monitor went to sleep and blacked out. I would see it happen. Suddenly I would be aware of it, and see it go to black. As if the change in circumstance was replayed. I did not know it was going to go to black. So was not paying attention to it as it happened. But yet I saw it happen. This happened regularly. I tried looking up some instance of visual stimuli being somehow replayed by the brain, but found no information.
But in the case I mentioned, there was not movement, any more so than movement that was happening in any other part of the room. My subconscious head turn had me making eye contact with someone who had been looking my way.
They were within your peripheral vision, and eyes turning towards you is still movement.
Indeed, one could probably make a good (though speculative) case that a lot of the reason for having good movement detection in our periphery is to notice being looked at. Quickly spotting a (potential) mate, friend or foe, would have had good selection pressure for our ancestors.
Prey animals really watch your eyes. Looking at dear, or horses, or anything, makes them really nervous. If you want to be gentle, you have to studiously avoid letting them see two eyes. People seem to be the same, perhaps it’s dominance thing: “You looking at me” is a meme in bar fights.
I am having trouble getting across what I am trying to ask. As I was walking, there was movement all around me. People working out, playing basketball, walking around. The one thing that got me to turn my head was eyeballs. Did my unconscious brain detect eyeballs better than my conscious brain could, and say, “Let’s see who’s looking at you!” Or was this just a random event that I am overthinking?
Our perception of the world is synthesised by our brains and has been shown to include things like this, where the narrative includes noticing things before they happened.
To misquote and badly-paraphrase something I heard someone else say recently: you’re not really looking at the world; you’re the audience at a show that your brain is putting on about it.
I believe it’s true that our visual system can detect patterns in the periphery of vision, not just movement.† And you can train yourself to be sensitive to them.
Anecdote ahead, but easily reproducible empirically:
I have at various times, collected different things that I found on the ground - I spent a whole year picking up lost or discarded change; I frequently go beachcombing for shells, fossils or seaglass; I’ve spent mornings picking wild strawberries.
All of these things can be a little difficult to spot at the beginning, but after some practice and repetition, something happens that is called ‘getting your eye in’ - your vision becomes trained to notice specific patterns and, often quite suddenly, they are no longer difficult to spot.
Given enough training, this process becomes less and less conscious and there comes a point where it’s almost as if someone else whispered ‘look over there’ - you look over there and you find an example of the sought object that you had not consciously looked for or noticed.
After a whole year of looking for dropped coins, it almost felt like a palpable and completely involuntary ‘tug’ on my eyeballs, diverting them to look at what turned out to be an example of the object I had been training myself to seek. I would spot coins from across the street, at night, without deliberately looking for them.
†. Peripheral vision is extra-sensitive to movement/change and there is a sense in which all vision is stimulated by movement or at least change in the stimulus reported by the receptors in the eye - indeed I think there have been experiments demonstrating that when a subject’s eyes are constrained from moving around (Saccades - which they do naturally and involuntarily), static objects in the field of view fade and become invisible.
The instance makes me think that there may be a kind of visual memory buffer. Maybe specialized to the peripheral vision area due to evolution favouring it. Main vision is processed immediate and discarded, but peripheral goes into a short term buffer that can be triggered. Could be an advantage to survival. I suspect it happens more often than I notice. The circumstance where I noticed it was during narrowed concentration without any extraneous visual or other distraction.
Definitely agree, reality passes through the sieve of our brain. One cannot truly know if they are experiencing the true reality. I disagree with definitely seeing things before they happen. I am not a believer in the various concepts of time actually being a thing. But quite accurately predicting the future by ones knowledge of past and present is a thing.
It’s possible that somebody was looking in your general direction, but didn’t focus in on you until your head turning attracted their attention.
Or you have a superpower far beyond that held by mortal beings. 
Actually visual process starts in your retina…so much so that the retina can be considered a part of the brain that has extended out to line the back of the eyeball.
As an example, this can provoke a visual reflex that causes you to look at movement outside of your central visual field. In particular cells in the retina called horizontal cells integrate signals to send them on to the optic nerve. These are one of at least 6 types of cells in the retina that process data from rods and cones.
Further process is done in the visual cortex, and only then do you become conscious of what your retina has already seen and at least partially processed.