Greased BB
 

On an old greased bottom bracket is the race with the flats always non-drive side?

NDS will have the adjustable cup.
DS has the fixed cup.

Italian BB's have RH threads on BOTH sides. Others have LH threads on DS.

as far as I know yes, I've never seen and adjustable drive side cup, they have always been fixed as pictured above.

Properly, yes - but as some have right-hand threads on both sides it's of course possible to install reversed.

Firstly, I'd hope modern bottom brackets are greased.... The term you're searching for is cup-and-cone, as opposed to the cartridge bottom brackets more commonly seen today.

Almost all bottom brackets have the fixed cup on the right-hand side. However, I think Trek made a few mountain bikes which came with a BB that had the fixed cup on the non-drive side, and a left-hand threaded lockring and adjustable cup on the drive side. Quite why, I'm not sure.

I also put an italian BB back together with the adjustable cup on the drive side once. The damn thing came loose within about 5 miles, since BBs with right-hand threads on both sides need the drive side cup to be extremely tight to stop it coming loose by precession, and I evidently couldn't torque the lockring down that hard.

Since this cup is clearly marked 1.37 x 24tpi, we know it's BSC with left threads on the right and right on the left. As long as the OP remembers the thread direction when removing it, there's little chance of going wrong when he puts it back together. (I used to say it was impossible to screw this up, but over the years a few determined people proved me wrong on that score).

Phil Wood's BBs have an adjustable DS cup. They're the cat's meow for odd-ball frames, IMO.

Not directly to the OP's question but the first generations of Shimano's cartridge BBs were installed from the left side. So that side was the "fixed" one. The right side held the retaining/centering ring. Andy.

That and floating designs like the Phil Wood is why I prefer to call them right and left cups.

Fixed cup drive side, and adjustable cup non-drive side. I posted prematurely, there was only one way to put it back together. Thanks for humoring me.