Palindromic Rows/Columns

This game is for 2 players.

Start by drawing an n-by-n grid on paper.
(Who would have guessed??)

The players take turns filling in empty squares of the grid, one square per move. Each player fills in floor(n^2/4) squares, so that there are a total of 2*floor(n^2/4) squares (about 1/2 of grid) filled in at game's end.

Player 1 gets as a score the number of grid-squares whose states need to be changed (from filled to unfilled, or vice versa) in order to make each ROW a palindrome.

Player 2 gets as a score the number of grid-squares whose states need to be changed in order to make each COLUMN a palindrome.

The player with the SMALLEST score wins.

{I know that the rules could have had Player 1 score with the number of squares needed to make the *columns* into palindromes, and Player 2 could have had the rows, then the *highest* score wins; but if each player attempts to determine their own score, then with my way each player has an incentive to be as efficient as possible in determining how many squares need to be changed to achieve the palindromes.}

Thanks,
Leroy Quet