A few possiblilities come to mind.
#1- if they installed the becket first, and then wound all the cable onto the drum, it would have pushed all the drum cable twist to the becket. They would have had all the cable laying out on the floor, off of the cable spool, to do that when they installed it. I don't know why they would have done it that way, but its possible.
#2- wrong rope for the application- some rotation resistant ropes can't be used in multipart lines. They are meant for drum to a ball only, and the twisting from going through a block will give them core damage at the becket. It will happen in truck cranes over time when running block with a rotation resistant rope that isn't meant for multipart.
#3 they gave the becket 10 or 12 twists before pinning it up, to try to cure cable twisting the block up, but that's usually only necessary on running non rotation resistant rope with a lot of cable on a long boom. Looks like you are a overhead crane without a lot of fall, so I don't know why they would have turned it before installing.
Either way, you've probably got some core damage in that last 6' of cable. If there's extra cable on the drum that you could still get the block to the ground, you could remove 10' of cable and reinstall the becket and see if the problem persists, then you would know if its a installation issue, or the incorrect cable.
It does look like a left lay cable on a right to left feeding drum, so I don't think its the wrong cable lay.