coordinate freedom vs equivariance, again

With Soledad Villar (JHU) and others I have been discussing making generalizations (or restrictions?) of image convolution operators to make machine learning respect more symmetries. One kind of generalization is going to 3-d images, and another is making the weights in the convolution filters geometric objects, like vectors, pseudovectors, and tensors. Then we developed a group-averaging technique to make these geometric filters equivariant. And now we are considering products and contractions of these geometric objects to make universally approximating function spaces. I don't love the word “equivariant” here: In my view the symmetries are coordinate freedoms, not relations between inputs and outputs. But the machine-learning world has spoken.