I live in New York State, I am hired by my company and registers to the NY office. I am on an assignment in New Jersey and conduct physical work at our client locations in NJ and NY. Why am I paying NJ tax and NY tax... What if I am injured, which state reimburses workers comp? What if I am out of work, in which state do I file unemployment?
The basic rule is that you owe taxes to the state where you do the work. In some cases the states can ignore it if the number of hours is small, though each state can define small differently.
Other pairs of states have reciprocity agreements where you only pay the where you live. these pairs tend to be neighbors and it would be a nightmare to have every hour tracked for every employee. Talk to your HR/payroll to make sure that the taxes are being assigned correctly.
Regarding workers comp and unemployment. Those are arranged through your employer, and if claims are needed they can advise you.