I was reading the solution to an exercise, and I don't really understand the logic passages in this sentence ($\Sigma \subseteq \mathbb{R}^3$ denotes a surface, and in particular the surface $\{(x,y,z) \in \mathbb{R}^3 \mid x^2+2y^2+3z^2=6\}$, but I don't think it's relevant to the question)
The orthogonal reflections of $\mathbb{R}^3$ with respect to the coordinate planes can be viewed as isometries $\Sigma \longrightarrow \Sigma$, and this means that the points $( \pm1,\pm1,\pm1)$ have the same principal curvatures and Gaussian curvature
It is clear to me that two isometric surfaces have the same Gaussian curvature because of the Theorema Egregium, but what about the principal curvatures? Why we can conclude the same thing?