First(?) Grid Game

Many of my games, if not most of them, are based on a square grid.
This game might be the first such game I posted on-line.

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Let n be some positive integer. (The best{ie. most fun} n is unknown
to me.)
Each player gets one of two identical pieces of n-by-n
graph/tracing-paper. (The top of each piece is marked somehow.)
Each player draws a line, without showing the line-in-progress to the
opposing player, from any square to any other square -- moving from
square to square either up, down, left, or right; and visiting each
square at least once.
Now, after players have done this, the two pieces are placed over each
other, superimposed so the pieces of graph-paper are in alignment.
Scoring is as follows:
For player-A, add a point for every square where:
*A's line goes from (W)est to (E)ast;
and B's line goes from (N)orth to (S)outh.
*A's line goes from W, takes a turn, and exits at (N);
and B's line goes from E, takes a turn, and exits at (S).
*A's line goes from W to S;
and B's line goes from E to S.
*A's line goes from W to S;
and B's line goes from W to S, as well.
*A's line goes from W to N;
and B's line goes from E to N.
*All 180-degree rotations of the above.
*All lines are in any direction, but have the same exit/enter-points
as the above. (Any enter-point can be an exit-point, and vice-versa,
for both A's and B's lines.)
--
A point is added to player-B's score for the same cases as above,
except all directions are rotated +-90 degrees.
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A point is taken from each players score:
*For every additional time, over once, the player's line visits a
square.
*When a line made by the player takes a "U-turn" (enters a square,
then immeadiately leaves it).
So, with a U-turn, 2 points, at least, are actually taken from a
player's score:
One for the u-turn itself, and one for entering the adjacent square
twice.
Highest score wins.
This game must have some interesting strategies (possibly involving
the edge squares). But perhaps not.

Each player can start and end their line on any square of his/her
choosing.
An advanced variation: The line must start and stop on the same
square, creating a single loop.


No score change occurs for any other cases of entrance/exit points.

Thanks,
Leroy Quet