The final section of Chapter 18 in Brave New World describes how the helicopter pilots uncover John the Savage's body in the lighthouse:

Just under the crown of the arch dangled a pair of feet.

The pilots yell "Mr. Savage," and we see that John hangs  from the rafters in the arch.  His feet dangle like the needles of the compass and the like the spinning of the helicopter blades outside.  He hangs metaphorically like Christ, whose death saved others.  John's literary death is to awaken us, the readers: to save us from ourselves.

He swings one way, and then another.  Such was the duality of his character: Savage and Utopian; lover and loather; Christ-figure and sinner.  In the end, he has no compass to guide him, no mentor, no books, no father or mother, no home, no direction, no hope.  And so he martyrs himself:

Slowly, very slowly, like two unhurried compass needles, the feet turned towards the right; north, north-east, east, south-east, south, south-south-west; then paused, and, after a few seconds, turned as unhurriedly back towards the left. South-south-west, south, south-east, east. …

In the last paragraph of the novel, John the Savage has killed himself.  He has hung himself and we see his feet rotating.  The narrator says that his feet are moving like compass needles and gives the directions in which they are pointing.

In my opinion, this is meant to symbolize that John has ended up being lost.  His compass does not know which way to point.  Before coming to "civilization" he knew exactly where the good and the bad were.  He had a moral compass.  Now that he has been exposed to civilization, he is losing that compass and that is why he has to kill himself.

So I think his rotating feet and the directions are important because they tell us he had to kill himself because he was lost (in a figurative sense).