Juana, Anish, and Giselle are comparing their solutions to the inequality

To solve the inequality \(\frac{5}{6}x \leqslant -10\), we first need to isolate \(x\).

We can do this by multiplying both sides of the inequality by the reciprocal of \(\frac{5}{6}\). The reciprocal is \(\frac{6}{5}\). Since \(\frac{6}{5}\) is positive, we do not change the inequality sign.

Here’s the step:

\[
x \leqslant -10 \cdot \frac{6}{5}
\]

Calculating the right side:

\[
-10 \cdot \frac{6}{5} = -12
\]

Thus, we have:

\[
x \leqslant -12
\]

Now let's analyze the statements of Juana, Anish, and Giselle:

- **Juana's statement**: She says the inequality symbol must be reversed because there is a negative sign in the inequality. This is incorrect because the inequality symbol only reverses when multiplying or dividing both sides by a negative number, which is not the case here.

- **Anish's statement**: He mentions the Multiplication Property of Inequality or Division Property of Inequality must be applied. This is correct because we do need to use multiplication by a positive reciprocal, hence the inequality symbol remains the same.

- **Giselle's statement**: She says the inequality symbol must stay the same. This is also correct since we multiplied both sides by \(\frac{6}{5}\), which is positive.

From this analysis, Anish and Giselle are correct, while Juana is incorrect.

Thus, the correct combination of people is:

**Giselle and Anish**.