Most often this problem is due to a bad tooth. I am viewing your pic on my phone(so this could be my imagination) but I may be seeing a bent tooth on the minute wheel near the yoke. The thing to do would be to remove the yoke and carefully examine all the gear teeth related to winding. If no problems can be found then clean and grease these parts and put it back together. Then try setting the watch while watching it all with a loupe. I suspect you'll find the problem but let us know if not and someone will come up with something else.-Cort
Yes I agree with Tom - the sleeve depth is super critical on these.
You might take the movement out of the case and with a square in your hand see if you can get it to set correctly (with the square not pushed in far enough to move the pin at all) and then see if you can get it to wind (with the square pushed in however far is necessary). If both of those seem to work right, move on to trying to get the sleeve depth set so they work when in the case. If they don't work, don't mess with the case until they do. The sleeve threads in the case are the last thing you want to mess up.
Keyless works wear, and they can be a big pain. Best of luck to you!
Your watch appears to not have the cam spring locked in correctly, this is supposed to have tension put on it by the lip of the case when the movement is correctly installed into the correct case. You will see a small tab that protrudes from the edge of the dial plate not far from the "cam spring retaining screw". This tab is what should be pushed in by the lip of the case, this in turn exerts pressure on the "cam spring" which in-turn pushes on the cam setting plate, when this happens it engages the intermediate setting wheel correctly with the minute wheel. This design feature was for when timing the watch, so no drag was present on the gear train.
Why it sets only setting backwards is due to the wheel inertia which throws the "cam setting plate down towards the minute wheel, inertia is present when the backwards motion is instigated, when setting forward in the normal direction the "cam setting plate" rocks away from the minute wheel. Easily missed if you have not worked on this type of mechanism but for anyone that has worked on Elgin pendant set watches this should be easily diagnosed (basic troubleshooting 101), otherwise maybe they should not be playing with them. You must check the adjustment in the above description before proceeding with the Pendant Sleeve adjustments as described by Tom & Chris.