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What is the difference in translation/meaning between these two different grammatical constructions:
Er müsste es getan haben.
Er hätte es tun müssen.
Another related question: What happens with this meaning when Konjunktiv in these sentences is replaced with Präteritum/Perfekt?
Er musste es getan haben.
Er hat es tun müssen.
All four examples seem to have identical meaning to me, and I am sure they don’t.
Related to this question.
First some relevant background knowledge:
If müssen is used to indicate obligation or similar, it is possible to use the subjunctive II to indicate that whatever was obligatory was not actually done (or is not done/will not be done). If you so wish, the argument verb’s subjunctive is shifted to the main verb. More about this here.
When indicating obligation or similar, müssen does not take a past infinitive as an argument. You cannot change your past, and thus you can only be obliged to do something in the present or future.
(An exception from this is to indicate anteriority, e.g.: “Ich muss es bis morgen getan haben.” – “I must have done it by tomorrow.” As this has to be explicitly mentioned in context and it isn’t in your examples, I assume that this case does not apply.)
Like all modal verbs, müssen acquires an additional meaning in the present subjunctive II mood (Konjunktiv II der Gegenwart), namely to likely be or could.
In the present or preterite indicative (but not in the perfect or infinitive), müssen (like the English must) also can be used to indicate that something is conclusively the case, e.g.:
Deswegen muss er der Täter sein.
Therefore he must be the culprit.
Now to your examples:
Er müsste es getan haben.
Since we have a past infinitive, müssen cannot indicate obligation here, but only possibility (points 2 and 3). Thus, this means:
It is likely that he did it.
Note that the aspect of likelihood can apply to es getan haben as well as er depending on the context and emphasis. With no special emphasis it is the former. With emphasis on er, it is the latter (“It was likely him, who did it.”)
Er hätte es tun müssen.
Müssen in the subjunctive II of the past (Konjunktiv II der Vergangenheit), hence it cannot mean to likely be (point 3). Assuming no further context, the only explanation for the subjunctive II is point 1. Thus this means:
He had to do it (but he did not).
Note that further context may explain the subjunctive II:
Gut, dass sie es getan hat. Sonst hätte er es tun müssen.
It’s good that she did it. Otherwise he would have had to do it.
Er musste es getan haben.
Since this is a past infinitive, the only valid interpretation is point 4: Indicating that you arrived at the conclusion that he did it. Note that in this meaning, this is equivalent to:
Er muss es getan haben.
Er hat es tun müssen.
This is the most straightforward case. As there is no finite form of müssen and no subjunctive, müssen can only indicate obligation here:
He had to do it.
Ok lets take the literal translations to explain the different meaning:
Er müsste es getan haben
This one means: Probably he could have done it, but I am not sure, if he really did.
He was not necessarily told to do something here, it could have even been some action that someone did, and you are just not sure, if it was him.
Er hätte es tun müssen.
Here the emphasis lies on müssen: He had to do it, but probably he nevertheless did not.
And now to you other examples:
Er musste es getan haben.
This is usually combined with some reference to a point in time like:
He had to have done it before last Tuesday...
No level of uncertainty in this one.
And the last one:
Er hat es tun müssen.
Again: no level of uncertainty. This sentence just says: He had no choice, he had to do whatever he did.
