Are there any nice properties we can talk about after using PCA to transform data to a new basis?

Question

I'm not talking about "compressing" data here....assume we kept all the eigenvector basis after performing PCA, and then used them to transform our data into a new basis of the same dimension as the original. Are there any interesting properties between the points that might hold or not hold or change/stay the same? E.g., are the relative distances between points the same (but just scaled)?