When we say surface templates/spaces, we can mean two different things.

1. Registration target: A standard sphere and the alignment data associated with the sphere (e.g., sulc, curv).
2. Resampling target: A set of standard vertices on the standard sphere.

The registration target and resampling target are often the same for volumetric data, but they can be different for surface data. For example, after obtaining sphere.reg with FreeSurfer (i.e., using FreeSurfer's registration target), we can use it to resample data to different resampling targets (e.g., fsaverage, fsaverage5).

To me it's confusing when we use the same name for both registration target and resampling target, e.g., using fsaverage to refer to both FreeSurfer's registration target and its resampling target with icoorder7.

I think it's better to name the registration targets based on the group of participants used to generate the alignment data (e.g., fsavg40, conte69, hcp196), and name the resampling targets with density/resolution (e.g., fslr-32k, fsavg-ico32). Does it make sense to people?

Some thoughts related to our discussion today @oesteban @effigies. I guess what Oscar wants is transformation between registration targets, and what I want is transformation between resampling targets.