  We develop a formalism for constructing the Pontecorvo-Maki-Nakagawa-Sakata
(PMNS) matrix and neutrino masses using an expansion that originates when a
sequence of heavy right handed neutrinos are integrated out, assuming a seesaw
mechanism for the origin of neutrino masses. The expansion establishes
relationships between the structure of the PMNS matrix and the mass differences
of neutrinos, and allows symmetry implications for measured deviations from
tri-bimaximal form to be studied systematically. Our approach does not depend
on choosing the rotation between the weak and mass eigenstates of the charged
lepton fields to be diagonal. We comment on using this expansion to examine the
symmetry implications of the recent results from the Daya-Bay collaboration
reporting the discovery of a non zero value for theta_{13}, indicating a
deviation from tri-bimaximal form, with a significance of 5.2 sigma.
