The projection is the mathematical transformation from the curved surface of the Earth to a mathematical surface. You can have types of projections based on the mathematical surface (conical, cylindrical...), or based on the features they want to rescue from this transformation (conformal, equal-area...), but, sorry, I've never heard of a classification based on these "slices". Moreover, now that I think of it, even those projections we are familiar with have to be cut somewhere.