House of mirrors

Here's the puzzle, and here's an interactive I wrote in PaperJS (Javascript). This interactive allows you to play around with the puzzle much more efficiently than by printing out grids, coloring them with markers, and using a pen to draw your lines of symmetry.

Let's find an upper bound on the number of lines possible.

Let $$r$$ and $$c$$ be the number of rows and columns respectively. Then the number of red diagonal lines from top left to bottom right, is $$r + c - 1.$$ The number of black horizontal lines, is $$r + 1.$$ The number of red horizontal lines which are coincident with a black line, is $$r + 1 - 4 = r - 3.$$ The number of red horizontal lines that cut the middle of a square is $$r - 2.$$ Therefore, the total number of horizontal lines is

$$r - 3 + r - 2 = 2r - 5$$

Thus, the total number of red lines possible, in a grid containing $$r$$ rows and $$c$$ columns, is

\begin{align} & 2(r + c - 1) + 2r - 5 + 2c - 5 \\ =\,& 2r + 2c - 2 + 2r + 2c - 10 \\ =\,& 4r + 4c - 12 \end{align}