This is almost always caused by friction coming from the cable housings. The reason is that in one direction the shifter is pulling the cable through the housing and must overcome any friction, which could be caused by contamination, corrosion, tight angles, and/or fraying. In the other direction it's releasing cable and does not have these forces to overcome.
You can test whether this is the case. On most bikes this is done by shifting into the lowest gear, then, while not riding or turning the cranks, shift the right shifter into the highest position. This creates slack in the rear derailleur cable. Grab the rear housing loop and pull it out of the housing stop on the chainstay. Do the same with any other housing stops you can. Then try working the shifter back and forth to see whether the friction is gone. While you're doing this you can do a simple clean and lube of the cable, which there are many guides to here, and depending on the cable layout you may be able to visually inspect for issues. Afterwards you can use the above steps in reverse to put it back together.
If the test revealed the problem to be cable friction, but cleaning and lubrication doesn't fix it, replacing the cable and housing is needed.