Rigid Motions Hidden in a Finite-Difference Identity

Split the classical identity sum_{s=0}^{r} (-1)^s C(r,s) (x-s)^r = r! (Tepper's identity) into its even and odd terms, and read each term, for r=2k, as the square of an edge. Two boxes appear with the same diagonal for every x. One fixed rotation carries one box onto the other, so corresponding edges meet at constant angles, which are given in closed form. Their limits sqrt(2/pi) and 2/pi come from the probability of a balanced run of coin tosses, and the odd half of the identity is the Gauss quadrature of the even half (a classical fact). We then ask when every entry of such a rotation is rational, and answer with a square criterion for divided-difference identities with any integer shifts, with a parity obstruction for symmetric sets of 2k+1 shifts when k is odd, explicit rational examples and parametric families. The series itself is classical; the contribution is the geometry of its even/odd splitting.