I'm not exactly clear what your goal is but see if this answers the question:
+ theme(axis.text.x = element_text(angle =
360/(2*pi)*rev( seq( pi/14, 2*pi-pi/14, len=14))))
That would make the names "tangential" to the splits. If you wanted them "perpendicular" (as in the illustration in the linked answer) you just add pi/2 radians to the angle. (Didn't we all take geometry in high school?)
+ theme(axis.text.x = element_text(angle =
360/(2*pi)*rev( pi/2 + seq( pi/14, 2*pi-pi/14, len=14))))
(BTW: The data argument that you specified had an error. I changed "24" to "14".)
Let me know if the first 7 labels look right, but you want the ones on the LHS of the figure flipped.
So you want the bottom 6 rotated by pi radians (=pi*360/2*pi degrees):
+theme(axis.text.x = element_text(angle = 360/(2*pi)*rev(
seq(pi/14,2*pi-pi/14, len=14))+
360/(2*pi)*c( rep(0, 4),rep(pi,6), rep(0,4)) ))
# the rotation "back" of the lower items
(I really do not understand the decision to use degrees rather than radians.)