Invertible shift versus unavailable task identity

The source multiplies row vectors on the right by

As a column-vector map this is ${R_{\theta }}^{\mathsf{T}}$, a rotation by $-\theta$. The angle sign must therefore be stated. At the configured $45^{\circ }$, the centers move to axis locations and their original labels remain attached.

These transformed clean laws have Bayes accuracy one. A trained classifier's accuracy loss is lack of generalization to the shift, not unavoidable loss of label information. It is not, by itself, demonstrated cognitive flexibility.