To solve the product \(\frac{-5}{10} \times \frac{-4}{10} \times \frac{-3}{10} \times \cdots \times \frac{3}{10} \times \frac{4}{10} \times \frac{5}{10}\), we can break it down step by step.

First, let's rewrite the product in a more manageable form:

\[
\left(\frac{-5}{10}\right) \times \left(\frac{-4}{10}\right) \times \left(\frac{-3}{10}\right) \times \left(\frac{-2}{10}\right) \times \left(\frac{-1}{10}\right) \times \left(\frac{1}{10}\right) \times \left(\frac{2}{10}\right) \times \left(\frac{3}{10}\right) \times \left(\frac{4}{10}\right) \times \left(\frac{5}{10}\right)
\]

Next, let's separate the numerators and the denominators:

\[
\frac{-5 \times -4 \times -3 \times -2 \times -1 \times 1 \times 2 \times 3 \times 4 \times 5}{10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10}
\]

The denominator is:

\[
10^{10} = 10000000000
\]

Now, let's calculate the numerator. Notice that the product of the negative numbers will result in a positive number because there are an even number of negative factors (5 negatives):

\[
(-5) \times (-4) \times (-3) \times (-2) \times (-1) \times 1 \times 2 \times 3 \times 4 \times 5
\]

First, calculate the product of the absolute values:

\[
5 \times 4 \times 3 \times 2 \times 1 \times 1 \times 2 \times 3 \times 4 \times 5
\]

This is equivalent to:

\[
(5 \times 4 \times 3 \times 2 \times 1) \times (1 \times 2 \times 3 \times 4 \times 5)
\]

Each part is \(5!\) (5 factorial):

\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

So the numerator is:

\[
120 \times 120 = 14400
\]

Putting it all together, we get:

\[
\frac{14400}{10000000000} = \frac{14400}{10^{10}} = 1.44 \times 10^{-6}
\]

So the final result is:

\[
1.44 \times 10^{-6}
\]