If we have a covariance matrix $X$, what does $(X'X)^{-1}$ result in? As the title states. Is there any significance behind this matrix? Is there a name?
Does it give any information that is not present in the first matrix?

If $\mathbf{x}$ is a covariance matrix then $(\mathbf{x}^\text{T} \mathbf{x})^{-1}$ is just a big mess with no particularly useful meaning (so far as I'm aware).  However, you might be interested to note that if $\mathbf{x}$ is a design matrix composed of variables that have been centered (i.e., shifted to a sample mean of zero) then $(\mathbf{x}^\text{T} \mathbf{x})/(n-1)$ is the sample covariance matrix for those variables (and so $(\mathbf{x}^\text{T} \mathbf{x})^{-1}/(n-1)$ is the inverse sample covariance matrix).
Irrespective of all this, obviously the matrix $(\mathbf{x}^\text{T} \mathbf{x})^{-1}$ could never give information that is not in $\mathbf{x}$, since it is a function of this matrix (i.e., there exists a function $\mathbf{x} \mapsto (\mathbf{x}^\text{T} \mathbf{x})^{-1}$).
