If we got a non-singular matrix $A$, how is the operation $Ã = (A^T)^{-1}$ called? And why is a matrix $A$ orthogonal if $A = Ã$?

Question

This was just introduced in our Differential Geometry course, but no name was given to this.

In the end, it's just the inverse transpose matrix, but does it have its "own" name?

Also, I don't quite understand the reason why a matrix A is orthogonal if $A = Ã$

I mean, as far as I know, a matrix A is orthogonal if it's columns (or rows) are orthogonal to each other. But why is this (apparently) equal to the condition, that $A = Ã$?