Below is a transcript of a conversation with a knitter about a specific make and model of machine. She has done all the right things but still has a one directional issue. The identical answer would apply in almost all situations where selection and knitting works well in one direction but fails in the other.

QUESTION: My Brother KH-260 knitting machine selects needles to the D-position only when the carriage goes from right to left. When it goes from left to right, all the needles end up in the B-position. My husband and I watched all your videos on the subject. According to them, the timing is correct and the punch card unit moves and turns a punch card. We cleaned and oiled all the parts. Everything looks good, except it is not. Maybe you have an idea what is wrong with this machine.

ANSWER: This is a classic case of a common event. You cleaned well, but almost certainly, there remains a little bit of crud or corrosion on the post of one of the flippers where you can't see it. You can find out where the problem is by turning the carriage upside down. Go through the process of engaging and disengaging KC and all the buttons that affect selection. For other brands of machine, use all the controlling buttons and levers for that model. 

Both sides of the carriage should respond immediately, and wherever you spot either failure to respond or a sluggish or incomplete response is the trouble. Normally, a drop of Kroil at the base of the problem post will free it. After applying Kroil, work the flipper to and fro manually.  If it doesn't work right away, allow it to sit overnight and try again. Kroil creeps slowly into where it's needed. If this process doesn't do the job, it's possible a spring has become damaged or detached. Also, occasionally, it's necessary to disassemble the flipper and hand clean the post. These two operations  are generally reserved for professional mechanics. But usually, Kroil does the job.