(Perhaps off-topic) Isn't that the default? Doesn't every company do that -- pay you the bonus and then Uncle Sam (and friends) take a cut depending on the tax situation?
I checked one of my offer letter (not from google) and the wording is like "an amount of $XXXXX grossed up for U.S taxes".
I don't know about the US, but in Europe it was always marked as expenses refunded to me on the payslip, which is tax free.
A hiring bonus would presumably always be pre-tax, since there's no expenses to reimburse.
In the US, if they gave you $1000 straight-up for moving costs, you'd probably see something like:
1. Company gives you $1000.
2. $750 hits your bank account after withholding.
3. You spend $1000 to move.
4. You file taxes, claim $1000 deduction for moving expenses.
5. You receive $250 refund.