I'm changing my search from Negamax to Negascout. Should the evaluation be completely symmetric or can it also consider advantages (more points) from the current player's point of view?

Symmetry is not a requirement. There are games where having the initiative is worth a lot, and a good evaluation function should have a term for that.

Also consider that some games just aren't symmetrical (eg Fox and Hounds) and the evaluation will reflect that