What is the “value” of your right to castle?

How do you half lose your right to castle?

That doesn't answer how you can half lose it.  You either lose it or you don't.

The right to castle is worth about a pawn give or take. White being able to castle black not = 1.2. Black being able to castle white not = -0.65. Latest stockfish with contempt at zero. 

Komodo say -0.50 for black being able to castle, white not. 

Its value is relative to the possibilities to develop active play against the rival's King position on the board. This means that, sometimes, it's better not to castle on one side or in neither of them.

A second consideration is to connect (and coordinate) the heavy pieces, which is generally of value in most situations although more as a guideline than as a rule.

A third consideration is when to. For example, in some openings a side pushes Ph2–h3 or Ph7–h6, while the center is closed or stabilized. In those cases, it may be better to delay the 0-0 until the other side has King-castled, to avoid being exposed to a fast Pg7–g5–g4 or Pg2–g4–g5, which may open the g–file and ease an attack on the own King. Another situation occurs when some critical maneuvers are happening somewhere on the board and a tempo up (or down) makes the difference; in such case, it may be convenient to delay the castle until the situation is clarified.

Perhaps when you move one of your rooks and can't castle only on that side. #noenema

Castling rights themselves may not be valuable enough.  Seems to be more of tactical value rather than strategic.  So if you get a winning attack then good.   Otherwise no.  Sicilian najdorfs are played often without black castling. 

I'd give up my left knight to castle

Probably all players have a different rate. There are people that simply play with king at center everytime, others don't know how to play with king at center. For me, a pawn, maybe two at max.

To me, the value of every piece depends on the position.  

The lazyman's answer. I'd rather prefer elaboration than "it depends on the position" if everyone said that. 

That's how it is. It's sufficient for the question. Unnecessary extension is not needed. What elaboration do you want . Usually , early castling is considered bad for several reasons such as the opponent might castle opposite side and start pushing pawns on your castled side. Also, they can formulate king hunting strategy by conducting their forces to the king side. That's why one should wait till adequate development to attack the centre and defend the king from early attack. After early castling, thinking that their king is safe, some beginners  go for early king incursion  before hardly developing their forces to  sustain the attack. There are implied rules for sacrificing pieces for bringing the king out in the center. Some beginners think that just because the opponent hasn't castled yet, they  must sac and push the king out for attack. I disagree to that because the lack of castling is not enough for such attack. It is applicable, only if the opponent lacks both in development and castling (this happens when someone plays excessive pawn moves in the opening). So, even though castling renders king safety in most occasions and completes the development by connecting and activating the rooks,    in a position where most of the opposition pieces are active on a particular side, and the pawn structure of your other side is shattered and weak and kings residence is at the center,  castling will be a positional error because it endangers your king . Therefore, castling in that position has negative value.  Similar to that, value of other pieces varies from position to position . If you want elaborated explanation from me, then I will have to write an entire book on that. Mainly , there are pieces activities which get more priorities over inactive pieces. However, there are limits to this priority. These are known as positional imbalances, which  poses questions like material vs piece activity and etc . There are books and lessons about positional imbalances. You can try searching for them for further clarification . Silman's book contains a lesson on this.

wow, great post @ancient-bot

Thank you

"Unnecessary extension is not needed"

You cannot castle across check or into check. Therefore, the ability to castle can be lost temporarily when an opponent's piece, often a bishop, but possibly the Queen and less often a Rook, cover any of the castling squares. Until that piece is moved or blocked, castling to that side is not possible. And, as already noted a Rook move results in the permanent loss of castling ability to that side (50% loss) and a King move eliminates the ability to castle permanently.

I have no idea what all of this is worth, perhaps the game?