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?