SD is a measure of spread and in other words how close or distant the data are from the mean.

So, yes for SD to be same for -p,-q,-r as for a p,q,r.

We can check the same alternately by taking three values like 1,2,3 and calculating the SD which will be unaffected if we make the data points -1,-2,-3. This is largely because the formula analyses the distance of the points from the mean and a -5  will be "x" units away from the mean just like a +5. The distance is independent of the sign.

Even in the formula to calculate SD, you would find the distance squared as(x-Xmean)^2 in the numerator during calculations. This is because the distance from the mean is important and not the sign,