You mean, a sentence like I can't be bothered to go out now contains an apparently passive construction, while conveying an active meaning something like I won't bother to go out now? You could regard it as parallel to Latin deponent verbs, I suppose. I think I'd see bothered more as an adjective than a participle in such a context i.e. I am bothered being an equivalent construction to I am eager, rather than to I am forced (any of which could, more or less, be followed by to do X).

As regards the uniqueness or otherwise of bother, I suppose most verbs which can be both transitive and intransitive (with a reflexive sense) can be used in such ways. So, you can say I bothered him to do it last night (hence he was bothered (by me)) or I bother to do it all the time (hence I am bothered (by myself, presumably)). (I guess they have slightly different meanings actually; the transitive verb means "irritate, pester" or similar, while the intransitive one means something like "take the effort"). In a similar way, you could say The house was burned down last year with much the same meaning as The house burned down last year. Or I'm washed every morning by 8am, equivalent to I wash every morning by 8am. Or I am delighted to visit these gardens - I delight to visit... (rather archaic-sounding, perhaps, but involving a verbal form which can also be an adjective, like bothered). Or, hey, what about be worried? Very similar to bother eg. I am worried about opening this letter i.e. I worry about opening this letter.

I think it's that subtle change of meaning that qualifies it as a true deponent form, rather than a mere passive. It does seem to have a slightly reflexive quality, though, which might be a result of the English language's reluctance to have deponents running its sentences. This reluctance might have quite a lot to do with Old Norse, which quite frequently takes passive forms of verbs and makes them mean something completely different from the usual form if they're used deponently.

This is a wonderfully odd conversation. I don't get to argue about pointless bits of English grammar nearly enough. (Sits down, tries to parse previous horrible sentence; gives up.)

I just hesitate to describe verbs in English as passive or deponent, when it's constructions that carry the meaning, not individual inflected words as in Latin. It's like saying, in the sentence John eats the cakes, that cakes is an accusative plural. The category of accusative is meaningless for English nouns - we only think to regard the noun in that way by comparison with languages like Latin. (Some pronouns are a different matter eg. John eats them, where I would be happy to describe them as an accusative, or object form.) Of course, the cakes functions as the object of the verb, but this is conveyed purely by word order - by the construction. So I'm reluctant to say "be bothered" is a deponent verb; though I'm happy to regard the construction as undergoing a sort of grammaticalisation, whereby it may have originally been seen in an entirely passive sense - I can't be bothered to get up as implying purely No one can bother me to get up or perhaps I can't bother myself to get up - but now implies a more active sense akin to I won't bother to get up. But since the construction can still be understood in the passive or reflexive sense, or even as involving an adjective rather than a past participle, I don't find it helpful to assert that it is deponent, in the sense that one can quite unequivocally say that moriri is a deponent form.

With regard to Old Norse, when you say it quite frequently takes passive forms of verbs and makes them mean something completely different from the usual form if they're used deponently, I presume you're talking about the middle voice (forms ending in -sk, -mk, -zk etc.)? Yes, this can be used purely in a passive construction eg. landit eyddisk af, or with reflexive sense eg. broethr munu berjask, or to convey an active meaning utterly different from the active form of the verb eg. andask vs. anda! Historically they're all reflexives, though, since -sk comes from sik. I'd be dubious about direct influence on English, though - the middle forms of Old Norse have no parallel with Old or Middle English forms.

To correct myself, my example broethr munu berjask of course illustrates a reciprocal usage - "brothers will fight each other" - not a reflexive one (though many instances of reflexive use of the Old Norse middle voice also occur, obviously).

Of course, the word 'middle' was totally eluding me. And the example given above, I can assure you, comes from one of the verses quoted in Snorra Edda. If I could remember which poem I'd be even more impressed.
The Norse analogy came to mind because of the closeness of the sense of 'be bothered' to a reflexive. My terminology was heavily influenced by Latin (which is what I read nowadays). It would be fun if one had a spurious Norse analogy though - I wonder how one could construct a plausible(-ish) transmission for it?