*not a complete answer, but too long for a comment:*

some experimentation$^*$ suggests that the diagonal entries of the diagonal matrix $S_n$ in the integer decomposition $M_n=U_nS_nV_n$, with $U_n,V_n$ unimodular, have the following form for $n> 2$:

$${\rm diag}\,S_n=1,1,2,2,2,\cdots 2,2,2,\{Z_n\},c_n$$

where the $\cdots$ indicate padding to length $n$ with a string of $2$'s and $\{Z_n\}$ is a string of integers given by
$$\{Z_n\}=\emptyset\;\;{\rm for}\;\;n\leq 5,$$
$$\{Z_n\}=\{6\},\{30\},\{6\},\{6,30\},\{6,30\}\;\;{\rm for}\;\;n=6,7,8,9,10,$$
$$\{Z_n\}=\{6,30\},\{6,30,210\},\{6,6,210\},\{6,30,210\},\{6,6,30,6930\}\;\;{\rm for}\;\;n=11,12,13,14,15,$$
$$\{Z_n\}=\{6,6,30,630\},\{6,30,30,630\},\{6,6,6,30,6930\},\{6,6,30,210,6930\},\{6,6,30,210,6930\}\;\;{\rm for}\;\;n=16,17,18,19,20,$$
$$\{Z_n\}=\{6,6,6,30,210,90090\},\{6,6,30,30,210,90090\},\{6,6,6,30,210,1531530\},\{6,6,6,30,30,630,90090\},\{6,6,6,30,30,630,90090\}\;\;{\rm for}\;\;n=21,22,23,24,25,$$
and so on. Unfortunately, I have been unable to detect a pattern in this sequence.

The final diagonal entry of $S_n$ is
$$c_n=\frac{2^{3-n}|{\rm det}\,M_n|}{\prod_{i}\tfrac{1}{2}(Z_n)_i}$$ following from the formula OEIS A023999 for the determinant of a spiral matrix:
$$|{\rm det}\,M_n|={\rm det}\, S_n=(3n-1) \frac{ (2n-3)!}{(n-2)!}$$

$^*$ if you would like to experiment further with spiral matrices, here are a few lines of relevant Mathematica code; I am intrigued by this $Z_n$ pattern, what is the logic behind it?