I am trying to understand the difference between a "congruence" or "similarity" transformation for two $n \times n$ matrices (which for the sake of simplicity, we'll assume are real). From what I can glean, a similarity transformation represents a change of basis from one orthogonal basis in $\mathbf{R}^n$ to another. My understanding is that a congruence transformation is an isometry, and so, it seems it would represent some geometrical operation like a (rigid) rotation, reflection, etc which preserves angles ad distances (but not necessarily orientation).
If someone can tell me if this is correct or correct any mistakes in my interpretation, I'd be most appreciative.
Thanks in advance.