If your character were struggling with his conscience, I'd argue for a single alignment. With an actual split personality, meaning that the orc side takes over every so often (triggers and rules are important, here*), your character could absolutely have multiple alignments. Something to look into is how the mind works when different personalities take over: Your good self wouldn't know what happened when they came around unless someone told them. They would only know what happened from the moment their persona switched back on. You would have the same physique, but a completely different psyche (memory, moods, methods, everything) for each character.
* Triggers for Switching:
Switches can be triggered by a wide range of
stimuli. Cognitive stimuli, such as depressive trains of thought
(Seligman, 1975; Beck, 1976), anniversary reactions (Hilgard & Newman,
1969), social situations or expectations may lead to state changes.
Self-induced volitional state changes probably occur in most
individuals. Bunney et al. (1972) found that discussions of passes or
discharge planning often appeared to trigger switches into mania.
Sensory stimuli are powerful triggers of state change, particularly
for evoking dissociative or anxiety states. Researchers studying
dissociative and anxiety reactions in posttraumatic stress disorder
make use of auditory and/or visual stimuli that are similar to combat
sights and sounds to evoke abreactions and anxiety reactions
(Dobbs & Wilson, 1960; Kolb, 1987), though cognitive imagery
techniques can be. equally effective (Pitman et al., 1987). Olfactory
(Kline & Rausch, 1985) and tactile stimuli can likewise be powerful
cues for triggering a switch into a dissociative or anxiety state.
(“Switch Process in Multiple Personality Disorder
and Other State-Change Disorders”,
Frank W. Putnam, M.D.
That being said, despite what a lot of the answers here are saying about alignment being unimportant, there are a more than a few items in the game that require a specific alignment to use, and/or have disastrous reactions when held by people of an opposing alignment. The knowledge is purely meta, but the good and evil tomes and talismans can do a solid number on you if you're holding them while you are the opposing alignment.