  We present a complete set of new flavour-permutation-symmetric mixing
observables. We give expressions for these "plaquette invariants", both in
terms of the mixing matrix elements alone, and in terms of manifestly
Jarlskog-invariant functions of fermion mass matrices. While these quantities
are unconstrained in the Standard Model, we point out that remarkably, in the
case of leptonic mixing, the values of most of them are consistent with zero,
corresponding to certain phenomenological symmetries. We give examples of their
application to the flavour-symmetric description of both lepton and quark
mixings, showing for the first time how to construct explicitly weak-basis
invariant constraints on the mass matrices, for a number of phenomenologically
valid mixing ansatze.
