  In this paper we work out in detail a new proposal to define rigorously a
sector of loop quantum gravity at the diffeomorphism invariant level
corresponding to homogeneous and isotropic cosmologies, and propose how to
compare in detail the physics of this sector with that of loop quantum
cosmology. The key technical steps we have completed are (a) to formulate
conditions for homogeneity and isotropy in a diffeomorphism covariant way on
the classical phase space of general relativity, and (b) to translate these
conditions consistently using well-understood techniques to loop quantum
gravity. To impose the symmetry at the quantum level, on both the connection
and its conjugate momentum, the method used necessarily has similiarities to
the Gupta-Bleuler method of quantizing the electromagnetic field. Lastly, a
strategy for embedding states of loop quantum cosmology into this new
homogeneous isotropic sector, and using this embedding to compare the physics,
is presented.
