Are measures of central tendency invariant to translation?
No, measures of central tendency get translated the same amount that the data values themselves do, so they are not invariant. Measures of spread are invariant to translation. They don't change when all the values are translated by the same amount.
For example, if your data set consists of the numbers
On the other hand, the (sample) standard deviation for both data sets is about 3.05 and the range for both data sets is 7 (both measures of spread).