My 6 geared Shimano bike doesn't shift from higher to lower gears but does the opposite smoothly
Separating the derailleur and shifter by undoing the cable is the correct diagnostic process. You now know the derailleur articulates freely, and the shifter shifts freely also. 

Next step would be to pull the cable at the derailleur end while working the shifter to see if tension produces friction or binding in the cable and housing.

Also closely inspect the cable run. Make sure that the housing is inserted into frame bosses properly. Look for kinks in the housing, runs of housing that are too short or anything where the cable might bind up  Pull the cable out and check for kinks or fraying.