Interesting question. There are several paradoxes for calcite. First of all, it should be noted that the correct space group of calcite uses the rhombohedral (trigonal) Bravais structural lattice (SG R-3c: R), which is quite impractical and therefore hexagonal projection is commonly used. Crystallographic structural plane (10-14), resp. shortly (104) corresponds in morphological crystallography to the faces of the basic rhombohedron (10-11), (101). This is due, among other things, to significant differences in the lattice parameters a and c. This faces also correspond to the cleavage of calcite and the preferential orientation in XRD.
I do not know at all that calcite preferentially grows on the substrate by the surfaces of the rhomboedron (104) or (101). I don't even know about the literature about this. On geological, resp. mineralogical samples it is not visible at all and I've seen a lot of them. The growth of rhomboedral faces is somewhat preferred kinetically. The chemical composition of the mother liquor also has an effect.
calcite crystal having 4 crystallography axes no 3 crystallography axes, and rhombohedry lattic is the better lattic among other in example cubic. ortorhombic, tetragonal and etc.