Irving says that he has been in MDR for three years; yet according to his employee badge, he has been at Lumon for nine years. Is it possible that he was severed before his current stint in MDR, and worked in another department? What if (see below) he was in Optics and Design, a place he seems infinitely more fit for, due to his skill in painting and his deep love for Kier lore, and there he fell in love with Burt and had to be …wiped? Re-severed? on the Testing Floor and moved to a new department to start again.
It has been theorized that Irving could be the artist who painted both The Grim Barbarity of Optics and Design and The Macrodata Refinement Calamity. The painting style of these two nearly identical works of propaganda is similar to Goya, a more wild and emotional painter than the more formal classical style of the other paintings distributed by O&D. His preferred tool at home is a palette knife, and the two paintngs in question share that rougher style. It is possible that Irving, before his severance procedure, painted these paintings. He could have, at the time, be reinforcing the separation of departments that is preferred by Lumon management.
It is speculated that, as part of his investigative process, outie Irving is trying to connect to his innie's memories with the music and repetitive painting of the hallway to the testing floor elevator. He keeps his lights dim, like that hallways, and drinks coffee so he stays up so late that his innie dozes off at work. Is he trying to force reintegration using techniques that he may have learned in PTSD therapy, or in his investigative work, or he's just trying this now?
Is Irving's outie part of the Whole Mind Collective, working with Reghabi, or just investigating on his own after forming his own suspicions about what Lumon is doing? If indeed he has an investigative background in his pre-Lumon past, he would have the pattern recognition (also honed by his love of art and music) to read between the lines and determine that Lumon is shadier than they present themselves.
It is clear that Irving comes from military background, both his father and himself have served. He could have opted for the severance procedure due to being unable to work due to his PTSD. His age puts him at the older end of being able to serve in Operation: Desert Storm, a short but intense conflict. Many veterans with PTSD are assigned dogs as companions to reassure them that there is no one in the next room; his dog being named Radar could indicate he's a M*A*S*H fan (Radar was a little bit prescient) or that the dog literally is his safety radar for unfamiliar spaces. Art therapy is also a method of working through PTSD; his technique is bold and confident but clearly not formally trained. If some of the other theories on this page hold water, his capacity for subterfuge and working undercover would also come from that training. His zeal as a company man may not be feigned, but certainly we know his outie does not share those sentiments.
Irving's outie life is comfortable and casual clothing, but his innie life is as prim and old-fashioned as a man many years his senior. He dresses as the perfect company man, and indeed, his innie is a perfect company man…until he isn't. But for years, as the senior refiner, he has tried to lead by example and wants everyone to be perfectly compliant even on trivial matters as the removal of a group photo. Yet we know outie Irving is actively investigating Lumon's shenanigans in a way not consistent with his innie's genuine reverence for the Eagan family and its philosophy.
The Baird Creek Bandit is described as a duteous and militaristic burglar breaking into homes in Baird Creek (Manor and Townhomes) stealing one valuable thing and leaving no trace, also doing something minorly disruptive before leaving to call attention to the fact that he or she was there. It fit's Irving's investigative nature and vehement, innate distaste for those who break the rules to be the kind of burglar who could enter a house leaving no exculpatory evidence, but then purposefully calling attention to it as a little “fuck you” to the person he has victimized. The items he steals may or may not be relevant themselves, but just a pretense to gain access to items in the home that he does not steal, so as to conceal his motives. I mean, it's just SO IRVING.