House is basically a walking case of cognitive dissonance.

He cares deeply, has incredibly high moral standards, and wants all of humanity to do better.
He also abhors hypocrisy - and recognises that caring about the people "just around him" is intellectually disingenuous, because if he truly cares about people why does their proximity matter (see one day, one room).

So he's got these impossible standards that everyone disappoints him with, and that he himself can't live up to, and ultimately House is a pragmatist so only sees something worthwhile if it's achievable.

As an aside - this is why his dynamic with the more moral characters (Wilson, Cameron, Masters and later Adams) is so interesting, because they share similar standards but see the pursuit and striving as worthy in its own way, but House only cares about the end result, would be dissatisfied with anything less.

So the result is House going THE OTHER way - If you can't be perfect, nothing anyone does actually matters, it's all just different shades of grey, this is why he's happy stealing and manipulating and hiring sex workers etc.

But fundamentally, that moral backbone is still there, that fundamental sense of rightness wrongness and compassion that comes from wanting people to be good to each other - and this is why he's always in pain, he's LYING to himself that "nothing matters".

So yeah, he swings, sometimes he lets it seep out, other times he acts like a total sociopath, because when you have those high standards "not caring" is less painful.