Is there an equivalent of the "King's rule" for sum? For integrals, there exists a formula (the "King's rule") :
$\int_{a}^{b}f(x)\textrm{d}x=\int_{a}^{b}f(a+b-x)\textrm{d}x$
Does it exist a sum counterpart ? Something like :
$\sum\limits_{a}^bf(x)=\sum\limits_{a}^bf(a+b-x)$
If it exists, could you provide a proof and possibly the name of this property, or at least a link ?

Well it’s fairly standard to say: $$\sum_{j=k}^na_j=\sum_{j=k}^na_{n-j+k}$$
The proof is clear. Every summand on the LHS appears on the RHS and visa versa, with no repetitions. Finite summation is independent of the order.
