I have a code which is spitting out matrices of the form
$$\left(\matrix{0&a&-a\\a&+\gamma&0\\-a&0&-\gamma}\right)$$
It has trace $0$ and thus its eigenvalues are of the form $0,\pm\lambda$ so that they sum to $0$.
This pattern seems to hold for larger matrices, where we get eigenvalues of the form $\{\pm\lambda_1,\pm\lambda_2,\dots\}$ with an additional $0$ if it has odd dimensions
For example the $8\times8$ matrix
Or for a size of $8\times8$ we have
$$\left(\matrix{0&0&0&0&0&0&a&0\\0&0&-a&0&0&0&0&0\\0&-a&0&a&0&0&0&0\\0&0&a&0&0&-a&0&0\\0&0&0&0&0&0&0&0\\0&0&0&-a&0&0&0&0\\a&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0}\right)$$
gives eigenvalues $\lambda={\pm a,\pm\frac 1 2 a \left(1-\sqrt 5\right),\pm\frac 1 2 a \left(1+\sqrt 5\right),0,0}$
Or for an alternative size of $8\times8$ we have
$$\left(\matrix{0&a&-a&a&b&-a&0&-b\\a&c&0&b&d&0&0&0\\-a&0&-c&0&e&-b&-e&-d\\a&b&0&c&d&0&e&-e\\b&d&e&d&g&0&0&0\\-a&0&b&0&0&-c&0&-d\\0&0&-e&e&0&0&0&0\\-b&0&-d&-e&0&-d&0&-g}\right)$$
Edit: For simplicity the $8\times 8$ case could be reduced to the form
$$\left(\matrix{c&0&b&d&0&0\\0&-c&0&e&-b&-d\\b&0&c&d&0&-e\\d&e&d&g&0&0\\0&b&0&0&-c&-d\\0&-d&-e&0&-d&-g}\right)$$
and we still get eigenvalues of the form mentioned. For example seeing the variables to iterating integers.
What is the general form of these matrices, and why do we get this repeated $\pm$ pattern?
Is it just that any symmetric matrix with trace $0$ will give eigenvalues of this form? (No, proven by generating a random one and it doesn't hold!)
The condition for a matrix $A$ to have $-\lambda$ as an eigenvalue whenever $\lambda$ is an eienvalue (and with the same multiplicity) is that its characteristic polynomial should be an even or odd function, i.e. either all the nonzero coefficients are odd or all the nonzero coefficients are even.
A more "geometric" condition is that $M$ is similar to $-M$. Thus in your first example, if you switch the second and third rows and the second and third columns, i.e. conjugate with $\pmatrix{1 & 0 & 0\cr 0 & 0 & 1\cr 0 & 1 & 0\cr}$, you transform $M$ to $-M$. Your other examples don't seem to work with a permutation matrix, though.
EDIT: Omnomnomnom's $6 \times 6$ example works with conjugation by the antidiagonal matrix $$ \left[ \begin {array}{cccccc} 0&0&0&0&0&1\\ 0&0&0&0 &1&0\\ 0&0&0&-1&0&0\\ 0&0&1&0&0&0 \\ 0&-1&0&0&0&0\\ -1&0&0&0&0&0 \end {array} \right] $$