... and I have been both situations for longer and have seen tons and tons of them (*)... So?

Not so hypotheticals -- Heck the inputs that you want labelled could be rotation matrices. The desired output could be a rotation matrix. Generating more convenient features could be via a rotation matrix. Dimensionality reduction could be through a reduction matrix. Sparsity could be encouraged by proper use of rotation matrices. Shows up if you want to build in group theoretic invariance in your predictive model.

(*) If you consider Householders then even more