Edit: Jeff’s answer - replace the cable housing - fixed the problem. I made sure the highest and lowest alignments were correct, that the cogs lined up with the gears, and then replaced all cable housing for the rear gears. It’s now shifting as expected.
I have the following rear components:
Shimano ST-4500 shifter, Tiagra
Shimano RD-M591 rear derailleur
They’re both supposed to be 9 speed. The rear derailleur is long cage. For completeness, I have a 3x front chainring set and that works mostly Ok.
The problem is when I shift from one direction to another, it takes two clicks to make the change. If I’m shifting in the same direction, one click works after that. I’m kinda-sorta used to it, but it’s annoying when I want to change just one gear in the other direction. Sometimes, after shifting twice to change directions, it’ll shift spuriously one more in that direction and then I have to reverse it to get it where I want it. It’s like one of the components is meant for 10 speeds and the other is for 9, but the manuals say they’re both 9.
Any suggestions? I’ve seen elsewhere that a Shimano RD-M592 would be compatible with the ST-4500, but I’m not sure why the M591 wouldn’t be as well.
Oh, cables are new; components have been cleaned and seem to work, otherwise. The cable housing from the shifter is somewhat old, and could probably be replaced, but the cable seems to slide through it well enough.