This is a dumb question, but oddly enough I can't find the answer out there. So...

I've installed Tektro R539AG caliper brakes.

Everything's fine, except now I suspect I put the barrel adjuster's spring on the wrong side: it made sense to me to put it UNDER the black barrel, so that it forced the barrel to mate with the metal insert on top, but I now see images that (for reasons I can't figure out) seem to suggest it should go between the black barrel and insert on top. I've kludged together a picture (this is a Shimano derailleur barrel, which looks basically the same) --

:

Is it supposed to be on that (upper) side? If so, WHY? Lastly: does it matter? I don't want to take stuff apart if I don't have to.