Is there a special name or something for when a matrix is invariant under a change of basis, i.e.
$$XAX^{T} = A$$
I'm trying to find what properties $A$ or $X$ have but it's a little hard to search for!
(I can get more specific - $X$ has determinant $1$ and $A$ is symmetric.)
Note that since $A$ is symmetric, $X^TAX = A \iff XAX^T = A$ (we can take the transpose of both sides).
A nice way to summarize this property is to say that the transformation given by $X$ preserves the bilinear form given by $A$. That is, for any vectors $u$ and $v$, we have $$ (Xu)^TA(Xv) = u^TAv $$ For examples of where matrices like these are of interest, you might want to look into the indefinite orthogonal group and the symplectic matrices. By Sylvester's law of inertia, we can think of your situation in terms of some indefinite orthogonal group as long as $A$ is invertible.