[5] The "n-gonal" part of the name does not refer to faces here, but to two arrangements of each n vertices around an axis of n-fold symmetry.

Twisted trigonal, tetragonal, and hexagonal trapezohedra (with six, eight, and twelve twisted congruent kite faces) exist as crystals; in crystallography (describing the crystal habits of minerals), they are just called trigonal, tetragonal, and hexagonal trapezohedra.

[8] Crystal arrangements of atoms can repeat in space with trigonal and hexagonal trapezohedron cells.

This is not to be confused with the dodecagonal trapezohedron, which also has 24 congruent kite faces, but two order-12 apices (i.e. poles) and two rings of twelve order-3 vertices each.

This is not to be confused with the hexagonal trapezohedron, which also has 12 congruent kite faces,[8] but two order-6 apices (i.e. poles) and two rings of six order-3 vertices each.

One degree of freedom within symmetry from Dnd (order 4n) to Dn (order 2n) changes the congruent kites into congruent quadrilaterals with three edge lengths, called twisted kites, and the n-trapezohedron is called a twisted trapezohedron.

If the kites surrounding the two peaks are not twisted but are of two different shapes, the n-trapezohedron can only have Cnv (cyclic with vertical mirrors) symmetry, order 2n, and is called an unequal or asymmetric trapezohedron.