V.S. Ramachandran, the neurologist famous for describing phantom limbs, also studied people with anosognosia for left-side hemiplegia (translated out of moon language: people whose left sides were paralyzed but they didn't know it). The tests he performed on these individuals make it clear that anosognosiacs aren't lying or sublimating their knowledge of their paralysis -- they genuinely have no idea that they're impaired.

When asked to pick up a tray full of champagne flutes, for example, anosognosiac hemiplegics would pick it up from one end, upsetting the tray and breaking all the glassware; of course, people who know they only have use of one hand will pick up a loaded tray from a middle balancing point. Another test involved offering a small prize (say, a small box of candy) for performing a one-handed task, and a larger one (a big box) for a task requiring two hands -- for instance, tying a shoelace. Paralyzed people who knew they were paralyzed would, of course, choose the one-handed task, the only one they could possibly perform. Fifteen times out of sixteen, however, anosognosiac hemiplegics would attempt the task with the bigger payoff, then sit gamely performing one half of the action necessary for shoelace-tying, not getting particularly frustrated or surprised at their failure. Someone who knew on any level that he or she was paralyzed would be likely to go for the prize that could actually be won, not the impossible task. When the subjects finally gave up, they would cite something like fatigue, arthritis, or bad eyesight.

This storytelling (e.g. "I couldn't tie the lace because of my arthritis") is known as confabulation. It occurs in other neurological disorders, such as Korsakov's syndrome, and (more mildly and more ambiguously) in normal brains, but it is frequently associated with anosognosia. Anosognosiacs will confabulate to immense degrees, trying to smooth over the cognitive dissonance created by being infirm and not realizing it.

Ramachandran, whose brilliant use of low-tech experimental setups was groundbreaking in his phantom limb experiments, used a similar construction to plumb the depths of his patients' confabulation. He rigged up a box into which the patient placed his or her hand. The patient's hand, when in the box, was hidden, and the visible hand belonged to an associate (probably some long-suffering teaching assistant), but the setup was such that it appeared to the patient as though it was his or her own hand. Ramachandran would then give the patient instructions to move his or her hand, and the associate would remain still. When Ramachandran gave instructions to keep the hand still, the associate would move. Basically, the hand in the box, which appeared to be the patient's own hand, would always do the opposite of what the patient was told.

Reactions to this were fairly stunning. Some patients denied that the hand belonged to them -- fair, since in this case it didn't, but anosognosiac patients often deny ownership of their limbs, a condition known as misoplegia. (In his book The Man Who Mistook His Wife For A Hat, Oliver Sacks describes one misoplegic who habitually fell out of bed... it turns out he was pushing out a dead limb he found in his bed night after night, which was of course really his own quite alive left leg.) Others insisted that the hand was performing as requested, and in some cases they denied the existence of a hand at all. One woman swore up and down that it wasn't a hand in the box -- it was a wedding cake. With the understanding that these people are not consciously lying or in denial, the implications of this are very interesting. After all, though these are brain-damaged individuals, still, the act of confabulation comes so naturally, and is so fully believed -- no matter how little sense it made, the woman truly believed there was a wedding cake in the box. Is this a natural preservative function of the brain? How much confabulation, one wonders, are we capable of in daily life? What we know of eyewitness testimony, as just one example, suggests we're capable of more than we often think.

There is one interesting temporary cure for anosognosia: a squirt of very cold water in the left ear. Patients who receive this treatment, called a "cold caloric," not only become fully aware of their paralysis, but realize that they have been paralyzed the whole time. Suppose you had asked a patient to stand up, for instance, and she had said she was too tired. After a cold caloric, not only would she cite the real reason ("I can't stand up, I'm paralyzed"), but she would be able to tell you that this was also why she couldn't stand up before. The effects of the cold caloric last from fifteen seconds to two hours, after which the patient not only reverts to an anosognosiac state, but doesn't remember ever having been lucid about his or her paralysis.

What exactly does this mean? Not totally clear, but Ramachandran theorizes that anosognosiac hemiplegics actually completely ignore the left sides of their bodies, which are sending them incongruous information ("we can't move," when the patients' brains have no reason to believe that this is true). They override this dissonant data with information their brains have already cached. Ramachandran describes it thus:

As an analogy, think of a military general in a war room, planning for a battle. He would ordinarily collect evidence from a large number of scouts, add it all up, and arrive at a decision on what to do next. Let’s say the information from his scouts led him to call for an attack the next morning. But just before the scheduled attack, imagine one more scout arrives and tells him that he has underestimated the enemy’s strength by a quarter. Even though that new information contradicts everything else he has been told, the general cannot afford to rethink the whole decision. Instead, he simply ignores the report from the new scout or tells him to get in line with what everybody else is thinking. He may even tell him to lie in order to preserve troop morale.
The cold caloric wakes up the paralyzed side and forces them to pay attention to it. Not a wholly satisfactory explanation, but the phenomenon is mysterious at best.

Sources:
Shreve, James. "The Brain that Misplaced Its Body." Discover, May 1995
Moscovitch, Morris. "Confabulation." Memory Distortion, ed. Daniel L. Shachter. Cambridge, MA: Harvard University Press, 1995.
and of course my neurophilosophy class