I’ve never seen it in person and couldn’t find any detailed pictures that show these panels but here is an excerpt from the description of the chair from New Advent Catholic Encyclopedia:
“The oak frame was reinforced by the insertion of pieces of acacia, and new panels of the same wood were made for the front and sides. The front panel is adorned with square slabs of ivory, disposed in three rows of six each, representing the labors of Hercules. Some of these slabs are placed upside down and were not, apparently, made for the places they occupy. The ivory ornaments of the back, on the contrary, are well adapted to the form of the chair; they represent the combats of animals, of centaurs and of men.”
I wondering if these were part of the original chair or added later on. Couldn’t find any resource explaining why they were chosen to be included.