In a homological algebra problem I am in the situation that I have an invertible (over $\mathbb{Z}$) integer matrix $X$ and a permutation matrix $Y$ such that $N:=XY$ is a matrix with all eigenvalues equal to 1 or -1.

Question 1 : Does such a situation appear already in other situations/fields in algebra/combinatorics? Is there an interpretation for this or does it have a combinatorial meaning?

Question 2: Do (invertible integer) matrices with all eigenvalues equal to 1 or -1 have a name? Are they studied in the literature?

In a homological algebra problem I am in the situation that I have an invertible (over $\mathbb{Z}$) integer matrix $X$ and a permutation matrix $Y$ such that $N:=XY$ is a matrix with all eigenvalues equal to 1 or -1.

Question 1 : Does such a situation appear already in other situations/fields in algebra/combinatorics? Is there an interpretation for this or does it have a combinatorial meaning?

Question 2: Do (invertible integer) matrices with all eigenvalues equal to 1 or -1 have a name, or do matrices $X$ as above have a name? Are they studied in the literature?