company housekeeping

Too vast and vague.

I will advise you to go through any good book on 5S and adapt that to your company environment.

here's a broom and dustpan.

use them in good faith

If you mean sanitation practices for eg a production facility, the usual approach is to decide on a cleaning routine, and then keep a log of the performance of those duties, signed by the person who is designated the task. This can be printed off as a task sheet with space for the person to mark the time and date and check off, sign off that it's done.

Record keeping is useful if there's any problem, you can then find out why it happened and what to change to prevent it in the future. Inspectors love records and in many jurisdictions they are required.

There are also some housekeeping terms in English that are sometimes applied to a company, in a way that might get you in trouble: such as "laundering the money" or "cooking the books".... Housekeeping metaphors are applied to crimes, and also pecadillos, as in "sweeping something under the rug"... and should probably not be used to refer to the "accounting" or "bookkeeping" or management side of business.

The first person to ask questions on this topic, is the person driving the broom!