I'm afraid not.
In all seriousness, your DM could decide that switching specializations falls under class feature retraining, but... it's a pretty huge change.
The same way any class works if you multiclass. You stop gaining the benefits of this class, and any ability that keys off of your Paladin level stops progressing.That does make sense, though. How do multiple classes interact with multiclassing? I would love to see a Paladin/Deathknight.
Also, by default, a Paladin and Death Knight can't multiclass. Paladins are any Good, while Death Knights are any non-Good.