Pulling my hair out trying to even find the correct description of this, and a lot of searching has turned up nothing concrete. The centering nut on my Shimano 600EX single-pivots has a sort of knurled or splined surface rather than normal wrench flats, as in the picture here:

https://flic.kr/p/WGjKyZ

As I understand it, to lock the brake to the frame without binding up the caliper arms, I need to simultaneously tighten that splined nut and the 6mm hex key on the other side of the frame mount, while setting the splined nut to the caliper arms appropriately to leave them free.

Some kind of proprietary cone-wrench-alike seems to be in order, but what is it called? In the likely event I can't find one, has anyone had luck with another tool?

I could be wrong or just not remember correctly but I think you can still get some purchase on that with a cone wrench

That's holding the brake arms on the pivot bolt; in order to tighten the brake to a frame/fork, all you need to do is hold the arms and tighten the female nut with a hex key. Fine centering adjustment is done with the cross-tip screw on the brake arm (or simply manhandling it into place with the brake arms).

If you needed to disassemble the brake for some reason, a 12-point socket/wrench will remove that nut.

Hi,

I thought I had tried this when I first approached it, but I would see that in tightening the female nut, the splined nut would turn with it, tighter and tighter against the caliper arms. This seemed to be making one of the arms bind, resulting in only one arm moving when pulling the lever. It appeared that allowing the splined nut to go further and further to the left (in the picture) would tend to bind the caliper arms, removing the tiny amount of 'play' required for smooth arm movement.

No doubt I am missing something but that's what it looked like. I will try it again with your advice. Thanks both!

A couple of thoughts:

-the nut is supposed to be tight enough for the arms to move freely/not bind. Hold the bolt in place with a hex key in the front, while you adjust the rear with the nut.

-this adjustment was held in place with either a setscrew (check all around the circumference of the nut) or loctite.

Good luck solving your problem!