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.

Look at the bottom side of the barrel where it mates with the caliper. See some little nubs there? They keep the barrel from rotating and mis-adjusting itself while you ride. The spring pushes those little nubs against matching divots on the caliper. To adjust the brake you pull the barrel back against the spring before turning it.

Correct. You could simply leave out the spring and plastic knob, the problem with that and with what you have now with it being missassembled is there is nothing to hold the adjustment except perhaps for some friction. The function of the spring and knob is to hold the adjustment by engaging the nubs on the calipers, and the knob also happens to make it easier to turn the threaded barrel.

Thank you!

EDIT: Okay, this is odd. I just looked at the Tektro plastic barrels, and -- unlike the Shimano ones in the pic -- they don't have any nubs. Completely smooth. Unless there's something I can't see without removing them.

Short answer: The spring goes inside the black plastic barrel and the threaded adjuster shank goes through the spring and screws directly into the brake arm.

Edit: OK, the the adjustment is just held by friction and the spring is needed to produce the required tension. The spring still goes inside the plastic barrel.

Thanks all. Fixed. (Less onerous than I thought it would be.) Yeah, the barrel doesn't have nubs to prevent movement, but the logic seems to be that if the adjuster shank is unscrewed sufficiently, the plastic barrel has to be pulled up to engage it -- otherwise the barrel just turns freely. (This isn't particularly rigorous logic, if that's the case. Most people will probably have the shank screwed closer to the brake arm.)