Assume that a square, symmetric matrix $A$ can be factored into $A=LDL^T$ where $L$ is unit lower triangular and $D$ is diagonal. For indefinite $A$, $D$ may have $2x2$ blocks on the diagonal. How much information about the spectrum of $A$ can we obtain from $D$?

For example, it is known by Sylvester's law of matrix inertia that the inertia of $D$ is the same as that of $A$ (they have the same number of positive and negative eigenvalues). This is interesting, but I am wondering what other information is hidden in $D$.