Are measures of central tendency invariant to translation?

What you mean by "translation" is not explicitly clear. Why you'd want to translate a statistical function mathematically is another question.
However, assuming a mathematical translation about some coordinates (where applicable) is desired, the results would not affect the relative values of the central tendencies.

You may see that somewhat by definition. Whatever size or shape your data is, it will have some calculable "central tendency". OF course, that may have NO SIGNIFICANCE at all! Also, the specific values WILL change (that is what a 'translation' does), but there reference and relevance to the data set will not be altered.

As an example, "translating" a data set mathematically will result in different values for things like the standard deviation and "Z values", BUT those values will be consistent with the translated data in describing the central tendency of the data set.

ALWAYS remember - statistical methods are just tools. They do not provide "solutions", only more numbers. It is up to the human to determine the proper tools, their proper usage, and their relevant interpretation. "Manipulating" statistical methods and results is neither helpful to the practitioner nor ethical to the public.