Yes, it's possible. No, it is not advisable.
Any a/c that has roll authority can roll inverted.
The a/c isn't certified for aerobatics - in part, defined as rolls beyond 60 deg AoB.
It hasn't an inverted fuel system so once you achieved -ve 'g' the engine would stop.
NB: I'm referring to a roll around the a/c axis, not a barrel roll. In which case the engine won't stop but you still will have exceeded the a/c's certification.
Also the non-aerobatic seat belt may not be adequate to keep you in the seat. Rather awkward to retain control when you're squashed upside down against the canopy. Especially if those brilliant PA38 door latches should fail to stay... er .... latched.
[This message has been edited by Tinstaafl (edited 25 June 2001).]