You're reading into it too much. And that's saying something because this is already one of the most discussed and dissected scenes in film theory courses from the last half century.

However, there is some symbolism with the reflection--though I might suggest that "symbolism" is too strong a word.

Symbolism in movies only really works when it appears as a motif--a recurring composition, action, object, musical cue, or dialogue. Symbols require definition to hold meaning. Authors employ motif to signal the definition before the symbol is placed into the context which the author wants to draw meaning from.

So, where else in The Matrix do we see reflections? Easy, it's the next scene. There's a cracked mirror near him when he sits down to be extracted, it then repairs itself and tries to kill him.

Briefly--because there's literal libraries worth of chapters and essays written on this scene--how things are perceived--or reflected--in the Matrix is a recurring topic of exposition and philosophical musing--Residual Self Image and all that. Morpheus' glasses reflect that theme and foreshadow many layers of what's to come. That Neo reaches "into the reflection" of the glasses foreshadows both Neo's "physical" contact with the mirror and the thematic journey he is embarking on. That the mirror immediately tries to kill him exposits the hazards of pursuing truth and foreshadows the threat his own reflection--his own self image--poses to him.

But trying to draw meaning from right hand/screen left is reaching into your own construct rather than the film's. It's already thematically hefty. That the reflection operates normally on screen should signal a weight limit. Your own words signal your value of "personal" perspective--your residual self image--more than the pursuit of truth. That mirror is hazardous.