Matt,
The pivot and it's lock/release mechanism are built into the plastic frame, the plastic frame itself is part of the lock/release mechanism, you can not just simply cut the pivot and mount it on the other side, the other side is totally occupied by the big releasing button, lock/release interconnect and 3 big springs, you have no place to re-attach the pivot, even if you attach it there, you will have to lose the lock/release function.
I would say only the craziest DIY would try that
