Is there such a thing as backwards sigma? Is there a math function, similar to sigma, that can run down? For example instead of $\sum\limits_{i=1}^{10}i$ , something that adds from 10 to 1 (like a backwards run)...
 A: Try
$$ \sum_{i=1}^{10} (11 - i)
$$
There is no reason to introduce another symbol when a simple subtraction can do the work.
A: If $a, b > 0$ then you can say
$$\sum_{k=a}^b a_k = \sum_{k=0}^b a_k - \sum_{k=0}^{a-1} a_k. $$
For example:
\begin{align}
  \sum_{k=3}^7 k &= \sum_{k=0}^7 k - \sum_{k=0}^2 k\\[0.3cm]
    &= (0+1+2+3+4+5+6+7) - (0+1+2)\\[0.3cm]
    &= 3 + 4 + 5 + 6 + 7\\[0.3cm]
    &= 25
\end{align}
And:
\begin{align}
  \sum_{k=7}^3 k &= \sum_{k=0}^3 k - \sum_{k=0}^6 k\\[0.3cm]
    &= (0+1+2+3) - (0+1+2+3+4+5+6)\\[0.3cm]
    &= -(4 + 5 + 6)\\[0.3cm]
    &= -15
\end{align}
A: There is nothing wrong with writing
$$
\sum_{i=10}^1 i
$$
A: Sure:
$$\sum_{i=0}^{9}(10-i)=10+9+8+\ldots+1$$
A: The problem with your example is that addition is commutative, so it is not really useful to have a distinction for a sum from $1$ to $10$ or from $10$ to $1$.
However, your question makes sense in a noncommutative setting. Suppose for instance you have 10 matrices $A_1$, ..., $A_{10}$. Since the product of matrices is not in general commutative, the product $A_1 \dotsm A_{10}$ is in general different from $A_{10} \dotsm A_1$. In this case, you may consider writing $\prod_{i=1}^{10}A_i$ in the first case and $\prod_{i=10}^{1}A_i$ in the second case, but this is probably not the most satisfying solution.
A better solution is to consider a totally ordered finite set $(I, \leqslant)$ and to write $\prod_{i \in I}A_i$. The first case of my example can now be obtained by considering the set $\{1, \ldots, 10\}$ ordered by $1 \leqslant 2 \dotsm \leqslant 10$ and the second one by  considering the set $\{1, \ldots, 10\}$ ordered by $10 \leqslant 9 \dotsm \leqslant 1$.
A: You could use the following notation (here for matrix products):
\begin{align}
&\prod_{i=1}^{10} A_i = A_1 A_2 \dotsm A_{10} \\ \\
&\prod_1^{i=10} A_i = A_{10} A_9 \dotsm A_1
\end{align}
The first means "for $i$ from 1 up to 10".
The second means "for $i$ from 10 down to 1".
