Life In 19x19

Hi,
I was wondering what are the basic principles regarding sente and gote while resolving a ko.

For example, this position arose in a game of mine (I am Black, it is White to play) :
If White wins the ko in A, she can capture the triangled stones.
If Black wins the ko, the triangled stones are saved.

In the game, White, captured in A, I played the threat in F6, White captures the triangled stones. I was expecting to capture the 7 white stones in compensation, but...
I must immediately connect the 9 black stones in F19 instead... And White can answer the ko threat !

The point is that White can win the ko in sente, while Black can only win the ko in gote.

Are there known principles about that ? For example, here, I could not get anything in compensation for loosing the ko. Or could I ?
Is there a classification of kos including double sente, sente, reverse sente and double gote kos ?
What becomes of ko threats in any case ?

See https://senseis.xmp.net/?SenteKo

There are kos such that taking the ko may be sente and kos such that winning the ko may be sente.

On a global basis, it looks like other plays are so small that this ko can be played either way. Also, if Black wins the ko, it leaves a ko which White can take with sente.

Here are some variations that may be of interest.

Thank you for the variations, Bill !

I just came around the same kind of ko just now, so it reminded me of this old thread. Here is the new position I came across:
Before the ko starts, the black stones were alive and the white stones were dead.
Then, White played L13. I didn't see the point, so I played elsewhere. Then White started the ko with T17.

Now, here is the way I see it :
If Black wins the ko (Black T18, then Black S16), he can save the black stones and the white stones are still dead. The sequence is gote.
If White wins the ko with F16, the white stones become alive. The sequence is sente because Black must answer M12 to save his stones.

I have thought about this kind of ko, and here is my conclusion for the time being :

Let winning a ko be sente for White. Let winning the ko be a move (here white S16) gaining W1 points, and let the next white move (here L14) gain W2 points and be gote. (omg, I now realize that I only considered the case W1 < W2).

Let's consider ko threats against White that consists in two black moves, ending gote (example : Black E3, then Black E4). Let's suppose that playing the threat (E3) gains nothing, while the execution of the threat (E4) gains something.

Let's consider the ko open for White.

Case 1 : black cancels the sente continuation in gote (here, Black plays M12). Result : White gains W1 points in gote.
Case 2 : Black plays a ko threat whose gain would be between 0 and W1. Result : White gains W1 points in gote, and Black has wasted a ko threat.
Case 3 : Black plays a ko threat whose gain would be between W1 and W2. Result : White gains W1 points in gote, and Black has wasted a ko threat.
Case 4 : Black plays a ko threat whose gain would be between W2 and W1+W2. Result : White gains between 0 and W1 in gote. The ko threat was useful.
Case 5 : Black plays a ko threat whose gain would be superior to W1 + W2. Result : White must answer the threat.

Does it seem correct to you ?
I've not yet considered the case W2 < W1.

Without considering ko values and sente / gote, a conceptually simple evaluation uses the method of reading and counting: read the interesting variations until kos and threats are executed, compare the resulting counts and choose accordingly. (Experts also take into account the temperature but plain reading and counting is often good enough.)