  We revisit mixing sum rule relations in the lepton and quark sectors under
the assumption that the 1-3 elements of the flavour mixing matrices
($V^u_L,V^d_L,V^e_L,V^\nu_L$) are zero in the flavour basis. We consider the
exact relations resulting from the validity of this "zero 1-3 flavour mixing
hypothesis" and analyse their implications based on the current experimental
data, including effects from RG running. In particular, we analyse how the
existing precise measurement of $\theta_{13}^\mathrm{PMNS}$ allows to derive
predictions for $\theta_{23}^\mathrm{PMNS}$ in models with constrained
$\theta_{12}^\mathrm{e}$. As examples, we calculate the predictions for
$\theta_{23}^\mathrm{PMNS}$ which arise in classes of Pati-Salam models and
SU(5) GUTs that relate $\theta_{12}^\mathrm{e}$ to $\theta_{12}^\mathrm{d}$. We
also derive a novel "lepton phase sum rule", valid under the additional
assumption of small charged lepton mixing contributions. We furthermore point
out that, in the context of GUT flavour models, the quark and lepton CP
violating phases $\delta^\mathrm{CKM}$ and $\delta^\mathrm{PMNS}$ can both be
predicted from a single imaginary element in the mass matrices.
