What I've found is:
First Player If N Is Odd (non-misere): The player takes the center piece and symmetrically imitates every one of the opponent's moves.
Second Player If N Is Even (non-misere): Player copies opponent's moves symmetrically. You will eventually take the last piece and win.
But what should I do when N is even and I go first, or when I go second and N is odd?
My best guess is I should count amount of rectangles and alone "chips" and, based on that, make moves, but the problem is that I'm not sure that this will work and maybe you have a better strategy for this game.