It does not really matter exactly how the employer chooses to count working hours. However, the Fair Labor Standards Act (FLSA) requires, among other things, that the employee is at least paid the official minimum wage for each hour worked, and gets overtime pay for >40 hours per week. The Act also defines what counts as "hour worked" for these purposes.
The Fact Sheet # 79D from the U.S. Department of Labor gives more details about what counts as "hours worked" for domestic services under the FLSA.
The general rule is that all time during the shift counts, whether the employee is busy or not:
Generally, when an employee is “on duty” (that is they must be in the
home and prepared to provide services when required), they are
The only exception is that time spent sleeping may be deducted under certain circumstances:
In some circumstances, an employer may exclude up to eight hours an
employee spends sleeping at the worksite from the time for which an
employee must be paid.
The rules are different, depending on whether the employee permanently resides at the place of work ("Live-in employee), has shifts of >=24 hours or shifts of <24 hours.
In your case the shift is >=24 hours, so the rule is (see page three of the document): Up to eight hours of sleep time may be excluded from hours worked, provided that:
- sleeping facilities are available
- the employee can usually sleep at least five hours without disturbance
- the employee agrees to exclude sleep time
Even then, interruptions of sleep for work must be paid, and if the employee get <5 hours sleep during a given shift, no exclusion is possible.
So in summary:
For calculating hourly wage and overtime pay, the employer must count at least 16 hours for each 24 hour shift, and more if the employee cannot sleep for eight hours during the shift.
It probably does not matter that the employer calls it "12 hours worked", but your mother is entitled to at least "minium wage x 16" for each shift worked, and is entitled to overtime pay (at least 1.5 times minimum wage) if she works more than two shifts per week.
If the shift is ever reduced to less than 24 hours, even sleep time may not be excluded.