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See #42
The design of the Seal was approved by a resolution of the Continental Congress in 1782. The only specifications of "thirteen" were 13 vertical stripes of red and white, 13 arrows, and 13 stars "forming a constellation". The description of the seal is written in this difficult heraldic jargon (e.g. "paleways of thirteen pieces, argent and gules, ...") and it's easy to suspect that the American writing this was a mere dabbler compared to the experts of the College of Heralds in London, or that several heraldic artists working from this language could come up with several different looking results. The same document that contains this technical description also contains an explanation of the symbolism written by Charles Thomson, secretary to the Congress.
The first physical seal was worked up the same year, presumably under Thomson's supervision. From the very beginning, and consistently, the "constellation" of the 13 stars was in 1-4-3-4-1 pattern, producing something resembling the (Jewish) 6-pointed star; also from the beginning and consistently, the eagle was facing (the viewer's) left, toward the side with the olive branch.
This first seal of 1782 had the 13 arrows specified by Congress. But a replacement seal worked up in 1841 by an artist named Throop had the eagle clutching only 6 arrows and had the stripes on the shield wrong - the red stripes were obviously wider than the white stripes. This caused some complaint. Unfortunately these same errors were perpetuated on a replacement seal worked up in 1877 by an artist named Baumgarten. Another thing: the first (1782) seal had the 13 stars each six pointed, but Throop made them all five pointed stars and thereafter they stayed five-pointed. So far the representation in the seal had been an attempt at being "realistic" - although the various artists were probably very unfamiliar with the bald eagle, they attempts to show a real bird (fairly thin and with a long neck, sort of like a wild turkey) attempting to fly while holding a brank in one claw, arrows in the other claw and a substantial shield hanging from its neck like a sandwich sign; the effect was not very inspiring.
In 1883 the government contacted the NY firm of Tiffany & Company for the design and production of a new seal. The work was primarily done by Tiffany's chief designer, British-born James Horton Whitehouse. Whitehouse took this project very seriously, doing a great deal of research and soliciting advice from a lot of people. Whitehouse was advised to use a "heraldic" rather than a realistic eagle. He also got conflicting advice on the design of the arrows - previously they had been classical European-type arrows with barbed points (just as appear now), but several people advocated using American Indian type arrows with sharpened stone points (he stayed with the classical arrows). He lowered the image a bit - now the eagle's throat is at the center of the circle and the clouds around the stars form a complete ring - previously the shield on the eagle's chest was the center and the clouds were jammed up against the top edge of the design so that they didn't form a complete ring. He restored the full count of 13 arrows, and also made sure that the olive branch had 13 leaves and 13 olives. When the Tiffany seal had to be replaced in 1904 the engraver, Max Zeitler of Baily Banks & Biddle (Philadelphia), did a very attentive job of reproducing the Tiffany artwork exactly and it remains unchanged to this day.
The "reverse" of the seal - the side with the unfinished pyramid - has a less clear history. It was described in the 1782 resolution but apparently, to this day, never actually engraved as a seal. The image of an unfinished pyramid appears in a $50 bill issued by the Continental Congress during the War for Independence, in 1778 ... in that image the pyramid is not so much building blocks as a series of 13 thick floors, and there seems to be a tiny doorway in the middle of the bottom floor; the pyramid is apparently out of doors but the ground is simply dark and flat. A drawing for the proposed seal in 1782 shows this pyramid towering over a couple of leafy trees in the background. A drawing done in 1786 had the pyramid on a constructed flat platform without any discernible surroundings - and much the same for a medal struck in 1882. Artwork done in 1856 shows 13 levels of building blocks on a pedestal labelled "MDCCLXXVI", on what appears to be a grassy meadow. Apparently the Egyptian-type desert, with a few scraggy weeds here and there, dates possibly from 1885 and hasn't changed since. It may be worth noting that the proportions, angles, etc. of the pyramid do not match those of the Egyptian pyramids - nor I think of the Mayan or Aztec pyramids.