RoyJ writes:

Looks like an auto-centering center punch to spot the end of a shaft.
Appears from the photos that the outer sleeve and 4 angled blades
move together, and independent of the center rod.
Would work best with stock that has been turned square on the
end. Place flat against the end of the shaft, slide the
sleeve/blades down until it's centered up, smack the center rod to
spot for center drilling.

Which would make it perfect for centering a chunk of square stock for
a wood lathe. Beats marking it with a pencil and hoping you get the
point in the right spot between grain marks.

It should work with round stock as well i.e. finding the center.