What about:
Try as he might, he wasn't able move it.
Hmm, the more I look at this one, the more baffling it seems. ;-)
Actually, it's quite simply the subjunctive mood. This is one of those fossils, a relic of the much more extensive subjunctive mood that English used to possess: it is akin to "as it were".
Remember what it is that we're trying to prove or disprove - "might is the past tense of may".
As I have pointed out in a different thread, the problem here is that the terminology being used by various people is imprecise and thus open to misinterpretation.
If we are going to have any kind of debate whatsoever, we need to first agree on what the words we use actually mean.
So when you say we are trying to prove or disprove that "'might' is the past tense of 'may'", you have to clarify exactly what you mean by the term "past tense".
I think that many people in this debate are trying to use two different meanings of the term at the same time, and that is where the confusion comes in -- we're talking at cross-purposes with ourselves, let alone with each other.
In effect, you're trying to use both the popular definition -- that which tells you when an action takes place -- and the technical definition -- which describes the
form of a verb -- at the same time, and this is doomed to failure.
If what you mean is that "might" (and other modals) rarely, if ever, refer to completed actions in the past, you are probably 100% correct. But if what you mean is that "might" is not the past tense form of "may", then you're actually making life a lot more complicated that it already is.
Consider this example you quoted, riverkid:
He said that he was going to Tokyo.
This is an example of what you call "backshifting". You correctly state that the presence of "was" does not indicate that he actually went to Tokyo. However, what it does indicate is that the state described as "is going to Tokyo" is now in the past -- compare with:
He says that he is going to Tokyo tomorrow.
The use of the present tense -- even in reported speech -- indicates that the plan is still current, and if you ask him now he will confirm that yes, he is due to go to Tokyo.
Things do get a little more complicated when modal verbs are concerned, because modal verbs indicate possibility, permission, obligation and related concepts. But consider:
"I have made a cake."
He said he had made a cake.
Pretty simple transformation: we have taken a present tense and made it a past tense. Now consider this parallel example:
"I may make a cake."
He said he might make a cake.
Now, you may object and say that it's possible to say, "He said he may make a cake," but I would then point out that it's equally possible to say, "He said he has made a cake." The difference is that "has" is definite -- we believe him -- while "had" is not so confident -- we are distancing ourselves from this claim.
And it is this which, in reported speech, past tense forms are used to indicate: "I'm not saying this is true, this is just what was said."
If you really want to make life horribly complicated for people trying to come to terms with English grammar, you could invent a raft of tenses and moods to cover all eventualities and invent grand-sounding names for them. Imagine...
will do: future determinate ("I
will pass this test!")
will do: future predictive ("Cameron will be the next Prime Minister.")
will do: future immediate ("The phone's ringing." -- "I'll answer it!")
will do: first conditional ("If the sun shines, we'll go outside.")
will do: present habitual ("She will always sneeze at the most inappropriate of times.")
...and we haven't even got to "would" yet, let alone "may" and "might".
But a past tense form is just that -- a verb's form. It says nothing about the
function of that form, it just describes the form. Thus, in a sentence like, "The train arrives in fifty minutes," the verb is in the present tense, but it describes an event that is timetabled for the future.
In modern English, past tense forms may indicate actions completed in the past, or they may indicate the hypothetical nature of an action. Past tense forms of modal verbs indicate permissions, possibilities and obligations that existed in the past (which is not usually useful), or permissions, possibilities and obligations which are hypothetical, or less likely, in general.
This is fortunate, because it makes describing things like the transformation of direct speech into indirect speech a lot less complex. The rule is, if you're not 100% sure that the speaker is telling the truth, look for the inflected verb. If it is a present tense form, change it to the past tense form. Watch:
"I play football."
He said he played football. (play -> play)
"I am going on Friday."
She said she was going on Friday. (is -> was)
"I have been waiting since 4 o'clock."
He said he had been waiting since 4 o'clock. (have -> had)
"I will be there."
She said she would be there. (will -> would)
"I may have made a mistake."
He said he might have made a mistake. (may -> might)
See how neat this is? See how elegant? True, it's a lot more subtle than that in some cases, but it's a lot easier to teach that way. All you need to do is to separate form from function, and remember the golden rule:
"Past tense" does
not describe when an action takes place -- it describes what a verb looks like; how it is
used is a completely different question.
This is, incidentally, according to the best principles of descriptive grammar.