  A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2
if it is combinatorially regular but "fails geometric regularity by a factor of
2"; its combinatorial automorphism group is flag-transitive but its geometric
symmetry group has two flag orbits. The present paper, and its successor by the
first author, describe a complete classification of regular polyhedra of index
2 in 3-space. In particular, the present paper enumerates the regular polyhedra
of index 2 with vertices on two orbits under the symmetry group. The subsequent
paper will enumerate the regular polyhedra of index 2 with vertices on one
orbit under the symmetry group.
