We establish a 1-1 correspondence between Valiant's
character theory of matchgate/matchcircuit
and his signature theory of planar-matchgate/matchgrid,
thus unifying the two theories in expressibility.
Previously we had established a complete characterization
of general matchgates, in terms of a set of
useful Grassmann-Pl{\"u}cker identities.
With this correspondence,
we give a corresponding set of identities
which completely characterizes planar-matchgates
and their signatures.
Applying this characterization we prove some
negative results for holographic algorithms.
On the positive side, we also give a polynomial time algorithm
for a simultaneous node-edge deletion problem,
using holographic algorithms.
Finally we give characterizations of symmetric
signatures realizable in the Hadamard basis.