I thought it would be nice

A) Are there any verbs that are usually not backshifted, and the whole sentence is rearranged instead? I think you mentioned "should". For example, if someone says "I should see a doctor", the reported version would be "He said he had to see a doctor" or "He said he'd better see a doctor" or "He was thinking of going to the doctor"???
As I mentioned in the previous post, I got freaked out by changing "I should be studying" to "I think you should have been studying", and it through off my reasoning completely.  Actually, as I also mentioned in the previous post, you can backshift should -- but to itself -- not to should have, unless time has passed and there is now regret for not having done what one should have done -- but then that would be reporting a different thought..  You just can't change the thought!  It's "I should be studying" >> (I thought) you said you should be studying. (I thought you would be studying because you said you should be studying.) And you can change the verbs if you want.  That's still an option, and one that's frequently used.

B) Some structures are weird, because I can't rephrase them like I'm telling a story, as above. One of those structures is "would like", as in "I'd like an ice cream", or "She'd like another coke". What's the backshifted version of those? Maybe I need to change them, and say "She told me she wanted another coke"...

Same as for plain would.
I would like some cake. >> I said I would like some cake.
She'd like another coke.  >>  She said she'd like another coke.

Compare:
I would have liked some cake. (I didn't get any) >> I said I would have liked some cake.
She would've liked another coke. (She didn't get another.) >> She said she would've liked another coke.
[These are 'counterfactuals' expressing mild regret.]

C) In the above examples, #6 and #7 have "might", and "might" is backshift to itself, "might". I was wondering if the same happened with "could". If you replace "might" with "could" in #6 and #7, would that be ok too?  Yes, I mentioned that in the previous post before I even realized you had asked about it..  I know "could" can be the backshifted version of "can", but I'm actually worried about "could" when possibility is involved. Like in this one, to make it clearer:
- If it's nice, I could go for a walk tomorrow morning. ---> I met Kate. She (could?)¹ go for a walk the following day. She could have gone for a walk...² (<-- Careful! This is a thought of mine, not related to what she actually said!) She said she (could?) go for a walk the following day.<sup>3

1</sup>Here again you have that problem of a past-form modal usually used with present-time meaning trying to pose as a carrier of past-time meaning in a main clause.  And here again, you really have to bury it in a subordinate clause governed by a past-tense main clause to get that across.  I'd go with "She said she could go for a walk the following day" or "She was available to go for a walk ... " or "She had time to go for a walk ...".

²I understand the meaning, but I'm stumped as to why it doesn't sound exactly right. My brain keeps jumping back to a counterfactual interpretation.  It's like one of those cubes where you can see it two ways, and your brain keeps flipping back and forth between the two.    It would probably work in a complex real-life situation, though.  Somebody might just spontaneously generate it, and it would be clear to me from context which of the interpretations was meant.

³could sounds perfect in this one.  It's just a matter of could backshifting to itself -- and the past tense in the main clause makes everything come out all right.
_______
General remarks in the form of an aside about conditionals with can / could.

The backshifts are as discussed earlier.
I could go for a walk.  >> I said I could go for a walk.
She could have gone for a walk. >> She said that she could have gone for a walk.

But in conditionals, strictly, can is to will as could is to would.
If it is nice, I [can/will] go for a walk
If it [were/was] nice, I [could/would] go for a walk.
If it had been nice, I [could/would] have gone for a walk.

So you have what's called a 'mixed conditional' to start with.  (Are you trying to give me a headache? )
But here, context tells us that a past form is being used with present meaning, in effect
If it's nice, you have the choice of going for a walk.
________

I'm not sure what more you need or what more I can say about these, but feel free to make up more examples, and we'll go through them together and see what we can see.  (But bring aspirin next time!)

CJ

1. You said "If tomorrow is nice, I could go for a walk" is a mixed conditional? Yes. The 'unmixed' form is "If tomorrow is nice, I can go for a walk." But that "could" is not counterfactual, so I think those kinds of sentences are fine as they are.  Yes, it's OK.  Do you usually say "You could try post in another section, if you wanted"? I'd say "...if you want".   Either is OK, but "if you want" is more common.  I feel if I say "If tomorrow was nice, I could go for a walk" it sounds like I don't expect tomorrow to be nice. True.  Don't worry about this stuff.  I was just judging the type of conditional on the basis of the forms, not on the basis of the meanings -- as a sort of "by the way" remark.  That stuff about will/would being like can/could was just some theoretical stuff -- nothing you need to worry about for practical use.

(I recommend forgetting this 'mixed conditional' discussion.)

2. My theory must not be right at all. It seems that the tenses in reported speech have nothing to do with the tenses you can use in normal sentences, for example in main clauses.   You're right to see that there is a disconnect there.
The reactor [might/could] have exploded. <--- Meaning 1: It is possible that it exploded. I don't know if it did. (Yes.  This is the factual interpretation.  The result remains unknown.)  - Meaning 2: It was possible that it exploded, but it did not actually exploded   (Yes.  This is the counterfactual interpretation.  The result is known, and it is the reverse of what is stated as possible.)
He said the reactor might/could have exploded. <--- Uh-oh! It seems only the second meaning is possible now: it was possible that it exploded, but it did not actually exploded. EDIT: No, wait, not true. Read the following lines. 

Actually, there's a third interpretation, because we might not even know yet if it actually exploded.  The important thing is that the possibility was present, not whether the event actually occurred. (It doesn't absolutely have to be counterfactual.)
The reactor [might/could] have exploded <--- Meaning 3?  Meaning 2B?  It was possible for it to explode (then, maybe even soon).  It was possible that it would explode (at the time or soon after).  (Whether we know now if it did or didn't is not relevant.)  [I don't know if you'd like to call this a variant of one of the other two, or if you'd like to consider it a third usage pattern.]

The same two (three?) interpretations are still possible when reported..  The use of He said does not change that.

So the same should be true for all the others in reported speech. If "Someone says something could/might/should/would have happened", the truth is that that thing did not happen. No, wait. This is not true. Because the reported version of "The reactor might/could have exploded, let's call the technician and ask him if it did" is "He said the reactor might/could have exploded, and wanted to call the technician and ask him if it did", isn't it?  Yes.  This is the third interpretation, where anyone's knowledge of the result is irrelevant.  It's just that something had the potential of happening at the time.  The whole idea is from the point of view of the past -- a sort of future of the past, so our present knowledge of what finally happened is irrelevant.

So I think the best explanation is "In reported speech modals don't change, unless the situation is counterfactual (something did not happen) and in that case you use modal+have+past participle".
I don't completely understand this section, particularly why you want to make an exception for counterfactual statements.
So, what's wrong with "should have studied" if she didn't study? She says: "I'm not going to study, even though I should study for the exam. It's tomorrow." ---> "She said that she was not going to study. She said she should have studied because the exam was the following day".   Frankly, should is a headache!  I would have used the simple identity backshift.  She said she should study because the exam was the next day.   I would wait until she herself said, "Even though I should have studied, I didn't" before reporting, "She said she should have studied".  This would be another identity backshift.

[Note:  should and must usually work differently from could and might, so it is probably going to be impossible to apply principles learned about could or might to examples with should (or must).]

4. I don't think I understand much about how to use "might" or "could" with past meaning in non-subordinate clauses. In another thread, we said that "At that time, they might hang you for such a crime" is ok. Yes.  It's a common practice / habit / activity "at that time".  It's 'epochal', so to speak.  (I made up that term -- it takes place during some epoch of history, or can be thought of that way -- a "used to happen that way" situation. This pattern is less used than the other, coming up next.)  That's not in a subordinate clause, yet you understand it as past. So I don't understand why you said that "That day the reactor wasn't working well. It might explode" is not ok. (Not a common practice / habit / activity.  No "used to happen" idea here. Just a simple event of exploding.  That's the difference. This meaning is much more common than main-clause might or could mentioned above.) And even more confusing is the fact that you said it should be "That day the reactor wasn't working well. It might have exploded". (= It had the potential of exploding = It would have been possible for it to explode) I see that as a later thought, knowing that it wouldn't explode in the end. Not really.  Whether you know is not the point.  See the material above regarding a third interpretation of might/could have.

CJ

Thank you so much Jim,
I really learned a lot from your last posts! A lot of things I didn't know. For example, I didn't know of that third interpretation.

Basically, the reactor [might/could] have exploded can also have this meaning, which I thought I had to express with "might/could", not "might have /could have":
- It was possible that it exploded, and I don't know if it did, or it is not relevant anyway.

Here's the last problem, hopefully:
There's no difference between the forms in direct speech and the ones in reported speech, unless you're reporting something counterfactual (now you'll see what I mean by that "counterfactual"):

The reactor [might/could] have exploded, thank God it did not. ---> He said the reactor [might/could] have exploded, but it didn't. <-- Same

The reactor [might/could] have exploded, but I don't know if it did. ----> He said the reactor [might/could] have exploded, but he didn't know if it did. <--- Same

The reactor [might/could] explode, if I pushed this button. Dangerous, isn't it? ----> He said the reactor [might/could] have exploded (if he had pushed that button). <--- Counterfactual, not the same forms. You could also keep the same forms in some cases, it depends, the meaning would be a little different but sometimes it doesn't matter.

The reactor [might/could] explode soon, but let's keep calm. -----> He said the reactor [might/could] explode soon and to keep calm anyway. <-- Same

Now, the problem is that those forms are not the same in non-subordinate clauses!
- That day, the reactor [might/could] have exploded, and we were told to keep calm. And in the end... Booom! It was Mark who told us that the reactor [might/could] explode.
- That day, the reactor [might/could] have exploded, and we were told to keep calm. And in the end... nothing happened! It was Mark who told us that the reactor [might/could] explode.
- That day, the reactor [might/could] have exploded, and we were told to keep calm. And in the end... I don't even know what happened, 'cause I fled! It was Mark who told us that the reactor [might/could] explode.

As you see, those are part of reported speech, but you also mentioned other sentences where the forms "might/could" should be used instead of "might have/ could have":
There was a danger that the reactor might explode. <--- and not "might have exploded".
How do I didtinguish between these usages? I mean, is "might/could" only to be used in subordinate clauses, or is there something weird here too? Like this sentence...
Everybody was afraid because the reactor might explode. <--- if that only applies to subordinate clauses, then this is not a subordinate, and "might" is ok, not "might have"...


Thanks again, I hope this thread will come to an end soon...

Thanks again!
You are not going to believe this... I problably understand! Well, at least 1000% more than before.

CalifJim wrote:

You have a choice.  I would guess that the preference is for the version with the fewest number of words. Reported "second" conditional (no backshift from original):  He said (it was dangerous and that) the reactor [might/could] explode (i