The pentagon clock
Everything in the formula is decided by where n lands on a regular pentagon. Step n around the circle by 72° at a time: the vertex it lands on gives the sign of the correction, +0.8, −0.8 or 0. This is the Legendre symbol n5, and it is why the cosine form F8 and the angle form F10 give the same areas.
n =
The first hand points at 72n°, the second at 144n°. The difference of their shadows on the horizontal axis is always +√5/2, −√5/2 or 0 (a quadratic Gauss sum modulo 5). Dividing by √5/2 gives the Legendre symbol.
Why the palindrome
The pentagon is symmetric about its horizontal axis. Reflecting it sends the vertex for n to the vertex for −n, and the colours match: vertices 1 and 4 are both +, vertices 2 and 3 are both −. So (−n/5) = n5, hence P(−n) = P(n), and the residues read the same backwards. Combined with the period 25, this gives the mirror at n = 25 and the 51-term palindrome.
In number theory this is the fact that −1 is a square modulo 5 (since 2² = 4 ≡ −1).