Use "of" or "for" with Institute, Department, Office...?

When should which be used and what's the difference?

Department of XYZ or Department for XYZ

Institute of ABC or Institute for ABC

Federal Office of... or Federal Office for...

Is there any sort of rule for this?

The "problem" is that it's an ongoing debate in the Department (non-English-speaking country) on how to properly translate it into English.

I believe you are specifically asking this question to avoid a possible embarrassing translation.

example:

Institute for the Criminally Insane

Most likely means an institute which helps and/or houses the criminally insane

Institute of the Criminally Insane

Perhaps is an institute run by persons who are criminally insane and/or an institute which advances the status of the criminally insane.

The second example is admittedly humorous and is intended to exaggerate the difference in connotation. Your use of either 'of' or 'for' should surely be determined by the context of the actual name it is to be applied to. No doubt someone in your own department's debate has mentioned a similar example to the one I have given.

Whether one uses "of" or "for" depends on the semantics. If the subject is part of a whole, it is 'of' its genus. If the subject represents, acts for or advocates a part (the object) - it is 'for' its subset. So we may say the University of the West Indies is the Secretariat for Caribbean Conferences.