There is something important within the speaker. To repeat the last line of Section 49, it is something "steady and central."
Whitman might be describing his own death here. His body becomes cool and he sleeps a long time.
The world hangs or "swings" on some central mystery, and the speaker finds reassurance in this feeling.
There is not ultimate death or chaos in the universe. The universe has a form and all life is connected. Ultimately, there is happiness in the union of all things.
It's as if the speaker is getting in touch with mysticism in this section.