If someone works over 40 hours in one week, is it legal to cut their pay in half to provide them with over time pay?

This would be illegal, since it is a plain attempt to avoid the clear requirements of overtime law: that non-exempt (including all hourly) emloyees get time-and-half (a *higher* pay rate) for all hours worked past 40 in a week. Employers may not circumvent the statutory (created by Congress or state legislatures) law, like the Fair Labor Standards Act, or the regulations (like those put out by the Dept. of Labor) implementing them. Jane appears to have a wage-and-hour and overtime complaint, and should contact the Dept. of Labor; she may be entitled to back wages & overtime.
There are legal ways to control overtime. For example:
1) The employer could not allow overtime.
2) If overtime is necessary, the employer may permanently reduce hourly wages for *all* hours worked to compensate for the expectation that overtime will be paid. (What they can't do is only have wages go down when there is overtime in order to reduce pay that week; that is not changing the job's basic pay but is trying to circumvent actually paying a $22/hour person $33/hour for overtime.) Basically, if the want Jane to make, say $900 per week and expect that she'll have 10 hours of overtime/week, they need to set her basic hourly wage at $16.36/hour, so it works out to $900/week on average, given overtime.
3) If an employee works overtime week 1, the employer can reduce her hours week 2 so as to save money (obviously, this assumes they can get by with fewer hours week 2).
But they can't reduce pay only when there is overtime, to negate the overtime laws and regulations.