[THOUGHT] To solve the problem, we need to find the invertible matrices $P$ with integer entries such that $P^{-1} = P^T$ (orthogonal matrices) and $PM = MP$ (matrices that commute with $M$).

First, note that $P^{-1} = P^T$ implies $P$ must be an orthogonal matrix. For an orthogonal matrix $P$, we have $P^T P = I$, where $I$ is the identity matrix.

Next, the condition $PM = MP$ implies that $P$ must commute with the diagonal matrix $M$. For a matrix $P$ to commute with a diagonal matrix, $P$ must also be diagonal. This is because if $P$ has non-zero off-diagonal entries, multiplying $P$ with $M$ will not yield the same result as multiplying $M$ with $P$, unless those off-diagonal entries are zero.

Thus, $P$ must be a diagonal matrix of the form:

Since $P$ is orthogonal, we have:

This implies:

Thus, each $p_{ii}$ can be either $1$ or $-1$.

Since $P$ must be invertible, none of $p_{11}, p_{22},$ or $p_{33}$ can be zero. Therefore, each diagonal entry has exactly 2 choices: $1$ or $-1$.

The total number of such $P$ is:

We need the square of this number:

We will use Python to verify the number of such matrices. [/THOUGHT] [PYTHON]

Count the number of 3x3 diagonal matrices with entries in {1, -1}

count = 0 for p11 in [1, -1]: for p22 in [1, -1]: for p33 in [1, -1]: count += 1 print(count**2)

[THOUGHT] The Python code confirms that there are 8 such matrices $P$. The problem asks for the square of this number, which is $8^2 = 64$. [/THOUGHT] $\boxed{64}$