2 expanded

The problem is equivalent to classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C_2$ of order 2, or $\mathbb{Z}[C_2]$-modules on $\mathbb{Z}^n$. This group has exactly 3 isomorphism classes of indecomposable free $\mathbb{Z}$-modules:

(1) trivial

(2) sign representation

(3) 2-dimensional with matrix $\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}.$

Every $n$-dimensional $\mathbb{Z}[C_2]$-module is a direct sum of (1), (2), (3) with uniquely determined multiplicities. Thus any involution is conjugate over $\mathbb{Z}$ to a block diagonal matrix with blocks [1], [-1], $\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}$ whose sizes are uniquely determined.

1

The cyclic group $C_2$ of order 2 has exactly 3 isomorphism classes of indecomposable $\mathbb{Z}$-modules:

(1) trivial

(2) sign representation

(3) 2-dimensional with matrix $\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}.$

Every $n$-dimensional $\mathbb{Z}[C_2]$-module is a direct sum of (1), (2), (3) with uniquely determined multiplicities.