Encountered this question in Knuth et al's Concrete mathematics. The question is:
What does $\sum\limits_4^0$ mean?
I think it does not mean anything unless we assign it meaning. It's just notation that can mean whatever we want. Maybe a useful meaning would be $\sum\limits_0^4$. This is analogous to scanning a list backwards in programming languages. You can go from 0 to n or n to 0. Does this answer the question?
If one thinks that the additivity property $$ \sum_i^j+\sum_{j+1}^k=\sum_i^k $$ is pivotal, one is led to define $$ \sum_4^0=-\sum_1^3. $$ More generally, for every integer pair $(i,j)$, $$ \sum_{i+1}^j=-\sum_{j+1}^i. $$ Edit: The OP now edited their question to explain why, in their opinion, one should use the convention that $$ \sum_j^k=\sum_k^j $$ for every $(j,k)$. Naturally, this is another possible option but, as I tried to indicate, it does not serve the same purposes as the one above. Actually, if indeed one chooses this other convention, I would suggest to avoid altogether the notation $$ \sum_{i=4}^0x_i $$ and to replace it by something like $$ \sum_{i\in I}x_i, $$ where $I=\{4,3,2,1,0\}$, since then the identity $I=\{0,1,2,3,4\}$ makes the intended meaning unambiguous.