  We present a continuum theory of graphene treating on an equal footing both
homogeneous Cauchy-Born (CB) deformation, as well as the microscopic degrees of
freedom associated with the two sublattices. While our theory recovers all
extant results from homogeneous continuum theory, the Dirac-Weyl equation is
found to be augmented by new pseudo-gauge and chiral fields fundamentally
different from those that result from homogeneous deformation. We elucidate
three striking electronic consequences: (i) non-CB deformations allow for the
transport of valley polarized charge over arbitrarily long distances e.g. along
a designed ridge; (ii) the triaxial deformations required to generate an
approximately uniform magnetic field are unnecessary with non-CB deformation;
and finally (iii) the vanishing of the effects of a one dimensional corrugation
seen in \emph{ab-initio} calculation upon lattice relaxation are explained as a
compensation of CB and non-CB deformation.
