(I don't have a source, only my interpretation, and a hint from the 1996 script about the state of <PERSON>'s powers by the time of the chase. Given the compressed timeframe of the final script, I take this to be a conservative estimate of <PERSON>'s level of awareness by this point in the movie.)
All of <PERSON>'s powers flow from the same place: an ability, on some level, to perceive and interact with the raw, unfiltered code of the Matrix.
<PERSON>'s superhuman reflexes are not in fact superhumanly fast - they are a cheat. They come from sensing and responding directly to changes in the Matrix, without waiting for this information to be filtered through a process of slowly and sloppily encoding and interpreting a series of audio-visual cues. 295
(For gamers: it's essentially the same as the Psycho Mantis trick from Metal Gear Solid.)
As <PERSON>'s powers grow throughout the movie, as his reactions get faster and faster, all that is really happening is that <PERSON> is becoming more and more aware of the Matrix itself and less and less dependant on the audio-visual illusions it casts.
I believe that the "other left" scene is a nod to <PERSON>'s level of awareness by this point in the movie. It is contemporary to a scene in the 1996 script:
Looking over his shoulder, watching the mirrors, <PERSON> drives almost unconsciously. (1996 script)
In the scene as originally scripted, <PERSON> is driving to the exit by car, apparently without needing to look where he is going, because he has another way of sensing what is ahead of him: his awareness of the Matrix. Worried more about the pursuing agents, <PERSON>'s focus is behind him rather than in front.
My interpretation of the "other left" scene is that it turns this scenario around - instead of, in the original script, physically looking behind him and "sensing" in front, <PERSON> is (more comfortably) physically looking ahead of him as he runs, but is "sensing" behind him, looking for the pursuing agents.
Because his focus is still on the agents, when he hears "left" he moves to the "left" that he perceives most strongly. 295